Functions that compute some of the sequences in Sloane’s tables¶
EXAMPLES:
Type sloane.[tab] to see a list of the sequences that are defined.
sage: a = sloane.A000005; a
The integer sequence tau(n), which is the number of divisors of n.
sage: a(1)
1
sage: a(6)
4
sage: a(100)
9
>>> from sage.all import *
>>> a = sloane.A000005; a
The integer sequence tau(n), which is the number of divisors of n.
>>> a(Integer(1))
1
>>> a(Integer(6))
4
>>> a(Integer(100))
9
Type d._eval?? to see how the function that
computes an individual term of the sequence is implemented.
The input must be a positive integer:
sage: a(0)
Traceback (most recent call last):
...
ValueError: input n (=0) must be a positive integer
sage: a(1/3)
Traceback (most recent call last):
...
TypeError: input must be an int or Integer
>>> from sage.all import *
>>> a(Integer(0))
Traceback (most recent call last):
...
ValueError: input n (=0) must be a positive integer
>>> a(Integer(1)/Integer(3))
Traceback (most recent call last):
...
TypeError: input must be an int or Integer
You can also change how a sequence prints:
sage: a = sloane.A000005; a
The integer sequence tau(n), which is the number of divisors of n.
sage: a.rename('(..., tau(n), ...)')
sage: a
(..., tau(n), ...)
sage: a.reset_name()
sage: a
The integer sequence tau(n), which is the number of divisors of n.
>>> from sage.all import *
>>> a = sloane.A000005; a
The integer sequence tau(n), which is the number of divisors of n.
>>> a.rename('(..., tau(n), ...)')
>>> a
(..., tau(n), ...)
>>> a.reset_name()
>>> a
The integer sequence tau(n), which is the number of divisors of n.
See also
- If you want to get more informations relative to a sequence (references, links, examples, programs, …), you can use the On-Line Encyclopedia of Integer Sequences provided by the - OEISmodule.
- If you plan to do a lot of automatic searches for subsequences, you should consider installing - SloaneEncyclopedia, a local partial copy of the OEIS.
AUTHORS:
- William Stein: framework 
- Jaap Spies: most sequences 
- Nick Alexander: updated framework 
- class sage.combinat.sloane_functions.A000001[source]¶
- Bases: - SloaneSequence- Number of groups of order \(n\). - INPUT: - n– positive integer
 - OUTPUT: integer - EXAMPLES: - sage: a = sloane.A000001;a Number of groups of order n. sage: a(0) Traceback (most recent call last): ... ValueError: input n (=0) must be a positive integer sage: a(1) 1 sage: a(2) 1 sage: a(9) 2 sage: a.list(16) [1, 1, 1, 2, 1, 2, 1, 5, 2, 2, 1, 5, 1, 2, 1, 14] sage: a(60) 13 - >>> from sage.all import * >>> a = sloane.A000001;a Number of groups of order n. >>> a(Integer(0)) Traceback (most recent call last): ... ValueError: input n (=0) must be a positive integer >>> a(Integer(1)) 1 >>> a(Integer(2)) 1 >>> a(Integer(9)) 2 >>> a.list(Integer(16)) [1, 1, 1, 2, 1, 2, 1, 5, 2, 2, 1, 5, 1, 2, 1, 14] >>> a(Integer(60)) 13 - AUTHORS: - Jaap Spies (2007-02-04) 
 
- class sage.combinat.sloane_functions.A000004[source]¶
- Bases: - SloaneSequence- The zero sequence. - INPUT: - n– nonnegative integer
 - EXAMPLES: - sage: a = sloane.A000004; a The zero sequence. sage: a(1) 0 sage: a(2007) 0 sage: a.list(12) [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] - >>> from sage.all import * >>> a = sloane.A000004; a The zero sequence. >>> a(Integer(1)) 0 >>> a(Integer(2007)) 0 >>> a.list(Integer(12)) [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] - AUTHORS: - Jaap Spies (2006-12-10) 
 
- class sage.combinat.sloane_functions.A000005[source]¶
- Bases: - SloaneSequence- The sequence \(tau(n)\), which is the number of divisors of \(n\). - This sequence is also denoted \(d(n)\) (also called \(\tau(n)\) or \(\sigma_0(n)\)), the number of divisors of \(n\). - INPUT: - n– positive integer
 - EXAMPLES: - sage: d = sloane.A000005; d The integer sequence tau(n), which is the number of divisors of n. sage: d(1) 1 sage: d(6) 4 sage: d(51) 4 sage: d(100) 9 sage: d(0) Traceback (most recent call last): ... ValueError: input n (=0) must be a positive integer sage: d.list(10) [1, 2, 2, 3, 2, 4, 2, 4, 3, 4] - >>> from sage.all import * >>> d = sloane.A000005; d The integer sequence tau(n), which is the number of divisors of n. >>> d(Integer(1)) 1 >>> d(Integer(6)) 4 >>> d(Integer(51)) 4 >>> d(Integer(100)) 9 >>> d(Integer(0)) Traceback (most recent call last): ... ValueError: input n (=0) must be a positive integer >>> d.list(Integer(10)) [1, 2, 2, 3, 2, 4, 2, 4, 3, 4] - AUTHORS: - Jaap Spies (2006-12-10) 
- William Stein (2007-01-08) 
 
- class sage.combinat.sloane_functions.A000007[source]¶
- Bases: - SloaneSequence- The characteristic function of 0: \(a(n) = 0^n\). - INPUT: - n– nonnegative integer
 - OUTPUT: integer; function value - EXAMPLES: - sage: a = sloane.A000007;a The characteristic function of 0: a(n) = 0^n. sage: a(0) 1 sage: a(2) 0 sage: a(12) 0 sage: a.list(12) [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] - >>> from sage.all import * >>> a = sloane.A000007;a The characteristic function of 0: a(n) = 0^n. >>> a(Integer(0)) 1 >>> a(Integer(2)) 0 >>> a(Integer(12)) 0 >>> a.list(Integer(12)) [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] - AUTHORS: - Jaap Spies (2007-01-12) 
 
- class sage.combinat.sloane_functions.A000008[source]¶
- Bases: - SloaneSequence- Number of ways of making change for n cents using coins of 1, 2, 5, 10 cents. - INPUT: - n– nonnegative integer
 - OUTPUT: integer; function value - EXAMPLES: - sage: a = sloane.A000008;a Number of ways of making change for n cents using coins of 1, 2, 5, 10 cents. sage: a(0) 1 sage: a(1) 1 sage: a(13) 16 sage: a.list(14) [1, 1, 2, 2, 3, 4, 5, 6, 7, 8, 11, 12, 15, 16] - >>> from sage.all import * >>> a = sloane.A000008;a Number of ways of making change for n cents using coins of 1, 2, 5, 10 cents. >>> a(Integer(0)) 1 >>> a(Integer(1)) 1 >>> a(Integer(13)) 16 >>> a.list(Integer(14)) [1, 1, 2, 2, 3, 4, 5, 6, 7, 8, 11, 12, 15, 16] - AUTHOR: - Gaski (2009-05-29) 
 
 
- class sage.combinat.sloane_functions.A000009[source]¶
- Bases: - SloaneSequence- Number of partitions of \(n\) into odd parts. - INPUT: - n– nonnegative integer
 - OUTPUT: integer; function value - EXAMPLES: - sage: a = sloane.A000009;a Number of partitions of n into odd parts. sage: a(0) 1 sage: a(1) 1 sage: a(13) 18 sage: a.list(14) [1, 1, 1, 2, 2, 3, 4, 5, 6, 8, 10, 12, 15, 18] - >>> from sage.all import * >>> a = sloane.A000009;a Number of partitions of n into odd parts. >>> a(Integer(0)) 1 >>> a(Integer(1)) 1 >>> a(Integer(13)) 18 >>> a.list(Integer(14)) [1, 1, 1, 2, 2, 3, 4, 5, 6, 8, 10, 12, 15, 18] - AUTHOR: - Jaap Spies (2007-01-30) 
 
- class sage.combinat.sloane_functions.A000010[source]¶
- Bases: - SloaneSequence- The integer sequence A000010 is Euler’s totient function. - Number of positive integers \(i < n\) that are relative prime to \(n\). Number of totatives of \(n\). - Euler totient function \(\phi(n)\): count numbers \(n\) and prime to \(n\). - euler_phiis a standard Sage function implemented in PARI.- INPUT: - n– nonnegative integer
 - OUTPUT: integer; function value - EXAMPLES: - sage: a = sloane.A000010; a Euler's totient function sage: a(1) 1 sage: a(0) Traceback (most recent call last): ... ValueError: input n (=0) must be a positive integer sage: a(11) 10 sage: a.list(12) [1, 1, 2, 2, 4, 2, 6, 4, 6, 4, 10, 4] sage: a(1/3) Traceback (most recent call last): ... TypeError: input must be an int or Integer - >>> from sage.all import * >>> a = sloane.A000010; a Euler's totient function >>> a(Integer(1)) 1 >>> a(Integer(0)) Traceback (most recent call last): ... ValueError: input n (=0) must be a positive integer >>> a(Integer(11)) 10 >>> a.list(Integer(12)) [1, 1, 2, 2, 4, 2, 6, 4, 6, 4, 10, 4] >>> a(Integer(1)/Integer(3)) Traceback (most recent call last): ... TypeError: input must be an int or Integer - AUTHORS: - Jaap Spies (2007-01-12) 
 
- class sage.combinat.sloane_functions.A000012[source]¶
- Bases: - SloaneSequence- The all 1s sequence. - INPUT: - n– nonnegative integer
 - OUTPUT: integer; function value - EXAMPLES: - sage: a = sloane.A000012; a The all 1s sequence. sage: a(1) 1 sage: a(2007) 1 sage: a.list(12) [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1] - >>> from sage.all import * >>> a = sloane.A000012; a The all 1s sequence. >>> a(Integer(1)) 1 >>> a(Integer(2007)) 1 >>> a.list(Integer(12)) [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1] - AUTHORS: - Jaap Spies (2007-01-12) 
 
- class sage.combinat.sloane_functions.A000015[source]¶
- Bases: - SloaneSequence- Smallest prime power \(\geq n\) (where \(1\) is considered a prime power). - INPUT: - n– nonnegative integer
 - OUTPUT: integer; function value - EXAMPLES: - sage: a = sloane.A000015; a Smallest prime power >= n. sage: a(1) 1 sage: a(8) 8 sage: a(305) 307 sage: a(-4) Traceback (most recent call last): ... ValueError: input n (=-4) must be a positive integer sage: a.list(12) [1, 2, 3, 4, 5, 7, 7, 8, 9, 11, 11, 13] sage: a(0) Traceback (most recent call last): ... ValueError: input n (=0) must be a positive integer - >>> from sage.all import * >>> a = sloane.A000015; a Smallest prime power >= n. >>> a(Integer(1)) 1 >>> a(Integer(8)) 8 >>> a(Integer(305)) 307 >>> a(-Integer(4)) Traceback (most recent call last): ... ValueError: input n (=-4) must be a positive integer >>> a.list(Integer(12)) [1, 2, 3, 4, 5, 7, 7, 8, 9, 11, 11, 13] >>> a(Integer(0)) Traceback (most recent call last): ... ValueError: input n (=0) must be a positive integer - AUTHORS: - Jaap Spies (2007-01-18) 
 
- class sage.combinat.sloane_functions.A000016[source]¶
- Bases: - SloaneSequence- Sloane’s A000016. - INPUT: - n– nonnegative integer
 - OUTPUT: integer; function value - EXAMPLES: - sage: a = sloane.A000016; a Sloane's A000016. sage: a(1) 1 sage: a(0) 1 sage: a(8) 16 sage: a(75) 251859545753048193000 sage: a(-4) Traceback (most recent call last): ... ValueError: input n (=-4) must be an integer >= 0 sage: a.list(12) [1, 1, 1, 2, 2, 4, 6, 10, 16, 30, 52, 94] - >>> from sage.all import * >>> a = sloane.A000016; a Sloane's A000016. >>> a(Integer(1)) 1 >>> a(Integer(0)) 1 >>> a(Integer(8)) 16 >>> a(Integer(75)) 251859545753048193000 >>> a(-Integer(4)) Traceback (most recent call last): ... ValueError: input n (=-4) must be an integer >= 0 >>> a.list(Integer(12)) [1, 1, 1, 2, 2, 4, 6, 10, 16, 30, 52, 94] - AUTHORS: - Jaap Spies (2007-01-18) 
 
- class sage.combinat.sloane_functions.A000027[source]¶
- Bases: - SloaneSequence- The natural numbers. Also called the whole numbers, the counting numbers or the positive integers. - The following examples are tests of SloaneSequence more than A000027. - EXAMPLES: - sage: s = sloane.A000027; s The natural numbers. sage: s(10) 10 - >>> from sage.all import * >>> s = sloane.A000027; s The natural numbers. >>> s(Integer(10)) 10 - Index n is interpreted as _eval(n): - sage: s[10] 10 - >>> from sage.all import * >>> s[Integer(10)] 10 - Slices are interpreted with absolute offsets, so the following returns the terms of the sequence up to but not including the third term: - sage: s[:3] [1, 2] sage: s[3:6] [3, 4, 5] sage: s.list(5) [1, 2, 3, 4, 5] - >>> from sage.all import * >>> s[:Integer(3)] [1, 2] >>> s[Integer(3):Integer(6)] [3, 4, 5] >>> s.list(Integer(5)) [1, 2, 3, 4, 5] - link = 'http://oeis.org/classic/A000027'¶
 
- class sage.combinat.sloane_functions.A000030[source]¶
- Bases: - SloaneSequence- Initial digit of \(n\). - INPUT: - n– nonnegative integer
 - OUTPUT: integer; function value - EXAMPLES: - sage: a = sloane.A000030; a Initial digit of n sage: a(0) 0 sage: a(1) 1 sage: a(8) 8 sage: a(454) 4 sage: a(-4) Traceback (most recent call last): ... ValueError: input n (=-4) must be an integer >= 0 sage: a.list(12) [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 1] - >>> from sage.all import * >>> a = sloane.A000030; a Initial digit of n >>> a(Integer(0)) 0 >>> a(Integer(1)) 1 >>> a(Integer(8)) 8 >>> a(Integer(454)) 4 >>> a(-Integer(4)) Traceback (most recent call last): ... ValueError: input n (=-4) must be an integer >= 0 >>> a.list(Integer(12)) [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 1] - AUTHORS: - Jaap Spies (2007-01-18) 
 
- class sage.combinat.sloane_functions.A000032[source]¶
- Bases: - SloaneSequence- Lucas numbers (beginning at 2): \(L(n) = L(n-1) + L(n-2)\). - INPUT: - n– nonnegative integer
 - OUTPUT: integer; function value - EXAMPLES: - sage: a = sloane.A000032; a Lucas numbers (beginning at 2): L(n) = L(n-1) + L(n-2). sage: a(0) 2 sage: a(1) 1 sage: a(8) 47 sage: a(200) 627376215338105766356982006981782561278127 sage: a(-4) Traceback (most recent call last): ... ValueError: input n (=-4) must be an integer >= 0 sage: a.list(12) [2, 1, 3, 4, 7, 11, 18, 29, 47, 76, 123, 199] - >>> from sage.all import * >>> a = sloane.A000032; a Lucas numbers (beginning at 2): L(n) = L(n-1) + L(n-2). >>> a(Integer(0)) 2 >>> a(Integer(1)) 1 >>> a(Integer(8)) 47 >>> a(Integer(200)) 627376215338105766356982006981782561278127 >>> a(-Integer(4)) Traceback (most recent call last): ... ValueError: input n (=-4) must be an integer >= 0 >>> a.list(Integer(12)) [2, 1, 3, 4, 7, 11, 18, 29, 47, 76, 123, 199] - AUTHORS: - Jaap Spies (2007-01-18) 
 
- class sage.combinat.sloane_functions.A000035[source]¶
- Bases: - SloaneSequence- A simple periodic sequence. - INPUT: - n– nonnegative integer
 - OUTPUT: integer; function value - EXAMPLES: - sage: a = sloane.A000035;a A simple periodic sequence. sage: a(0.0) Traceback (most recent call last): ... TypeError: input must be an int or Integer sage: a(1) 1 sage: a(2) 0 sage: a(9) 1 sage: a.list(10) [0, 1, 0, 1, 0, 1, 0, 1, 0, 1] - >>> from sage.all import * >>> a = sloane.A000035;a A simple periodic sequence. >>> a(RealNumber('0.0')) Traceback (most recent call last): ... TypeError: input must be an int or Integer >>> a(Integer(1)) 1 >>> a(Integer(2)) 0 >>> a(Integer(9)) 1 >>> a.list(Integer(10)) [0, 1, 0, 1, 0, 1, 0, 1, 0, 1] - AUTHORS: - Jaap Spies (2007-02-02) 
 
- class sage.combinat.sloane_functions.A000040[source]¶
- Bases: - SloaneSequence- The prime numbers. - INPUT: - n– nonnegative integer
 - OUTPUT: integer; function value - EXAMPLES: - sage: a = sloane.A000040; a The prime numbers. sage: a(1) 2 sage: a(8) 19 sage: a(305) 2011 sage: a.list(12) [2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37] sage: a(0) Traceback (most recent call last): ... ValueError: input n (=0) must be a positive integer - >>> from sage.all import * >>> a = sloane.A000040; a The prime numbers. >>> a(Integer(1)) 2 >>> a(Integer(8)) 19 >>> a(Integer(305)) 2011 >>> a.list(Integer(12)) [2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37] >>> a(Integer(0)) Traceback (most recent call last): ... ValueError: input n (=0) must be a positive integer - AUTHORS: - Jaap Spies (2007-01-17) 
 
- class sage.combinat.sloane_functions.A000041[source]¶
- Bases: - SloaneSequence- \(a(n)\) = number of partitions of \(n\) (the partition numbers). - INPUT: - n– nonnegative integer
 - OUTPUT: integer; function value - EXAMPLES: - sage: a = sloane.A000041;a a(n) = number of partitions of n (the partition numbers). sage: a(0) 1 sage: a(2) 2 sage: a(8) 22 sage: a(200) 3972999029388 sage: a.list(9) [1, 1, 2, 3, 5, 7, 11, 15, 22] - >>> from sage.all import * >>> a = sloane.A000041;a a(n) = number of partitions of n (the partition numbers). >>> a(Integer(0)) 1 >>> a(Integer(2)) 2 >>> a(Integer(8)) 22 >>> a(Integer(200)) 3972999029388 >>> a.list(Integer(9)) [1, 1, 2, 3, 5, 7, 11, 15, 22] - AUTHORS: - Jaap Spies (2007-01-18) 
 
- class sage.combinat.sloane_functions.A000043[source]¶
- Bases: - SloaneSequence- Primes \(p\) such that \(2^p - 1\) is prime. \(2^p - 1\) is then called a Mersenne prime. - INPUT: - n– nonnegative integer
 - OUTPUT: integer; function value - EXAMPLES: - sage: a = sloane.A000043;a Primes p such that 2^p - 1 is prime. 2^p - 1 is then called a Mersenne prime. sage: a(1) 2 sage: a(2) 3 sage: a(39) 13466917 sage: a(40) Traceback (most recent call last): ... IndexError: list index out of range sage: a.list(12) [2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127] - >>> from sage.all import * >>> a = sloane.A000043;a Primes p such that 2^p - 1 is prime. 2^p - 1 is then called a Mersenne prime. >>> a(Integer(1)) 2 >>> a(Integer(2)) 3 >>> a(Integer(39)) 13466917 >>> a(Integer(40)) Traceback (most recent call last): ... IndexError: list index out of range >>> a.list(Integer(12)) [2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127] - AUTHORS: - Jaap Spies (2007-01-26) 
 
- class sage.combinat.sloane_functions.A000045[source]¶
- Bases: - SloaneSequence- Sequence of Fibonacci numbers, offset 0,4. - REFERENCES: - S. Plouffe, Project Gutenberg, The First 1001 Fibonacci Numbers, http://ibiblio.org/pub/docs/books/gutenberg/etext01/fbncc10.txt 
 - We have one more. Our first Fibonacci number is 0. - INPUT: - n– nonnegative integer
 - OUTPUT: integer; function value - EXAMPLES: - sage: a = sloane.A000045; a Fibonacci numbers with index n >= 0 sage: a(0) 0 sage: a(1) 1 sage: a.list(12) [0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89] sage: a(1/3) Traceback (most recent call last): ... TypeError: input must be an int or Integer - >>> from sage.all import * >>> a = sloane.A000045; a Fibonacci numbers with index n >= 0 >>> a(Integer(0)) 0 >>> a(Integer(1)) 1 >>> a.list(Integer(12)) [0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89] >>> a(Integer(1)/Integer(3)) Traceback (most recent call last): ... TypeError: input must be an int or Integer - AUTHORS: - Jaap Spies (2007-01-13) 
 - fib()[source]¶
- Return a generator over all Fibonacci numbers, starting with 0. - EXAMPLES: - sage: it = sloane.A000045.fib() sage: [next(it) for i in range(10)] [0, 1, 1, 2, 3, 5, 8, 13, 21, 34] - >>> from sage.all import * >>> it = sloane.A000045.fib() >>> [next(it) for i in range(Integer(10))] [0, 1, 1, 2, 3, 5, 8, 13, 21, 34] 
 
- class sage.combinat.sloane_functions.A000069[source]¶
- Bases: - SloaneSequence- Odious numbers: odd number of 1s in binary expansion. - INPUT: - n– nonnegative integer
 - OUTPUT: integer; function value - EXAMPLES: - sage: a = sloane.A000069; a Odious numbers: odd number of 1s in binary expansion. sage: a(0) 1 sage: a(2) 4 sage: a.list(9) [1, 2, 4, 7, 8, 11, 13, 14, 16] - >>> from sage.all import * >>> a = sloane.A000069; a Odious numbers: odd number of 1s in binary expansion. >>> a(Integer(0)) 1 >>> a(Integer(2)) 4 >>> a.list(Integer(9)) [1, 2, 4, 7, 8, 11, 13, 14, 16] - AUTHORS: - Jaap Spies (2007-02-02) 
 
- class sage.combinat.sloane_functions.A000073[source]¶
- Bases: - SloaneSequence- Tribonacci numbers: a(n) = a(n-1) + a(n-2) + a(n-3). Starting with 0, 0, 1, … - INPUT: - n– nonnegative integer
 - OUTPUT: integer; function value - EXAMPLES: - sage: a = sloane.A000073;a Tribonacci numbers: a(n) = a(n-1) + a(n-2) + a(n-3). sage: a(0) 0 sage: a(1) 0 sage: a(2) 1 sage: a(11) 149 sage: a.list(12) [0, 0, 1, 1, 2, 4, 7, 13, 24, 44, 81, 149] - >>> from sage.all import * >>> a = sloane.A000073;a Tribonacci numbers: a(n) = a(n-1) + a(n-2) + a(n-3). >>> a(Integer(0)) 0 >>> a(Integer(1)) 0 >>> a(Integer(2)) 1 >>> a(Integer(11)) 149 >>> a.list(Integer(12)) [0, 0, 1, 1, 2, 4, 7, 13, 24, 44, 81, 149] - AUTHORS: - Jaap Spies (2007-01-19) 
 
- class sage.combinat.sloane_functions.A000079[source]¶
- Bases: - SloaneSequence- Powers of 2: \(a(n) = 2^n\). - INPUT: - n– nonnegative integer
 - OUTPUT: integer; function value - EXAMPLES: - sage: a = sloane.A000079;a Powers of 2: a(n) = 2^n. sage: a(0) 1 sage: a(2) 4 sage: a(8) 256 sage: a(100) 1267650600228229401496703205376 sage: a.list(9) [1, 2, 4, 8, 16, 32, 64, 128, 256] - >>> from sage.all import * >>> a = sloane.A000079;a Powers of 2: a(n) = 2^n. >>> a(Integer(0)) 1 >>> a(Integer(2)) 4 >>> a(Integer(8)) 256 >>> a(Integer(100)) 1267650600228229401496703205376 >>> a.list(Integer(9)) [1, 2, 4, 8, 16, 32, 64, 128, 256] - AUTHORS: - Jaap Spies (2007-01-18) 
 
- class sage.combinat.sloane_functions.A000085[source]¶
- Bases: - SloaneSequence- Number of self-inverse permutations on \(n\) letters, also known as involutions; number of Young tableaux with \(n\) cells. - INPUT: - n– nonnegative integer
 - OUTPUT: integer; function value - EXAMPLES: - sage: a = sloane.A000085;a Number of self-inverse permutations on n letters. sage: a(0) 1 sage: a(1) 1 sage: a(2) 2 sage: a(12) 140152 sage: a.list(13) [1, 1, 2, 4, 10, 26, 76, 232, 764, 2620, 9496, 35696, 140152] - >>> from sage.all import * >>> a = sloane.A000085;a Number of self-inverse permutations on n letters. >>> a(Integer(0)) 1 >>> a(Integer(1)) 1 >>> a(Integer(2)) 2 >>> a(Integer(12)) 140152 >>> a.list(Integer(13)) [1, 1, 2, 4, 10, 26, 76, 232, 764, 2620, 9496, 35696, 140152] - AUTHORS: - Jaap Spies (2007-02-03) 
 
- class sage.combinat.sloane_functions.A000100[source]¶
- Bases: - SloaneSequence- INPUT: - n– nonnegative integer
 - OUTPUT: integer; function value - EXAMPLES: - sage: a = sloane.A000100;a Number of compositions of n in which the maximum part size is 3. sage: a(0) 0 sage: a(1) 0 sage: a(2) 0 sage: a(3) 1 sage: a(11) 360 sage: a.list(12) [0, 0, 0, 1, 2, 5, 11, 23, 47, 94, 185, 360] - >>> from sage.all import * >>> a = sloane.A000100;a Number of compositions of n in which the maximum part size is 3. >>> a(Integer(0)) 0 >>> a(Integer(1)) 0 >>> a(Integer(2)) 0 >>> a(Integer(3)) 1 >>> a(Integer(11)) 360 >>> a.list(Integer(12)) [0, 0, 0, 1, 2, 5, 11, 23, 47, 94, 185, 360] - AUTHORS: - Jaap Spies (2007-01-26) 
 
- class sage.combinat.sloane_functions.A000108[source]¶
- Bases: - SloaneSequence- Catalan numbers: \(C_n = \frac{\binom{2n}{n}}{n+1} = \frac{(2n)!}{n!(n+1)!}\). - Also called Segner numbers. - INPUT: - n– nonnegative integer
 - OUTPUT: integer; function value - EXAMPLES: - sage: a = sloane.A000108;a Catalan numbers: C(n) = binomial(2n,n)/(n+1) = (2n)!/(n!(n+1)!). Also called Segner numbers. sage: a(0) 1 sage: a.offset 0 sage: a(8) 1430 sage: a(40) 2622127042276492108820 sage: a.list(9) [1, 1, 2, 5, 14, 42, 132, 429, 1430] - >>> from sage.all import * >>> a = sloane.A000108;a Catalan numbers: C(n) = binomial(2n,n)/(n+1) = (2n)!/(n!(n+1)!). Also called Segner numbers. >>> a(Integer(0)) 1 >>> a.offset 0 >>> a(Integer(8)) 1430 >>> a(Integer(40)) 2622127042276492108820 >>> a.list(Integer(9)) [1, 1, 2, 5, 14, 42, 132, 429, 1430] - AUTHORS: - Jaap Spies (2007-01-12) 
 
- class sage.combinat.sloane_functions.A000110[source]¶
- Bases: - ExponentialNumbers- The sequence of Bell numbers. - The Bell number \(B_n\) counts the number of ways to put \(n\) distinguishable things into indistinguishable boxes such that no box is empty. - Let \(S(n, k)\) denote the Stirling number of the second kind. Then \[B_n = \sum{k=0}^{n} S(n, k).\]- INPUT: - n– nonnegative integer
 - OUTPUT: integer; \(B_n\) - EXAMPLES: - sage: a = sloane.A000110; a Sequence of Bell numbers sage: a.offset 0 sage: a(0) 1 sage: a(100) 47585391276764833658790768841387207826363669686825611466616334637559114497892442622672724044217756306953557882560751 sage: a.list(10) [1, 1, 2, 5, 15, 52, 203, 877, 4140, 21147] - >>> from sage.all import * >>> a = sloane.A000110; a Sequence of Bell numbers >>> a.offset 0 >>> a(Integer(0)) 1 >>> a(Integer(100)) 47585391276764833658790768841387207826363669686825611466616334637559114497892442622672724044217756306953557882560751 >>> a.list(Integer(10)) [1, 1, 2, 5, 15, 52, 203, 877, 4140, 21147] - AUTHORS: - Nick Alexander 
 
- class sage.combinat.sloane_functions.A000120[source]¶
- Bases: - SloaneSequence- 1s-counting sequence: number of 1s in binary expansion of \(n\). - INPUT: - n– nonnegative integer
 - OUTPUT: integer; function value - EXAMPLES: - sage: a = sloane.A000120;a 1s-counting sequence: number of 1s in binary expansion of n. sage: a(0) 0 sage: a(2) 1 sage: a(12) 2 sage: a.list(12) [0, 1, 1, 2, 1, 2, 2, 3, 1, 2, 2, 3] - >>> from sage.all import * >>> a = sloane.A000120;a 1s-counting sequence: number of 1s in binary expansion of n. >>> a(Integer(0)) 0 >>> a(Integer(2)) 1 >>> a(Integer(12)) 2 >>> a.list(Integer(12)) [0, 1, 1, 2, 1, 2, 2, 3, 1, 2, 2, 3] - AUTHORS: - Jaap Spies (2007-01-26) 
 
- class sage.combinat.sloane_functions.A000124[source]¶
- Bases: - SloaneSequence- Central polygonal numbers (the Lazy Caterer’s sequence): \(n(n+1)/2 + 1\). - Or, maximal number of pieces formed when slicing a pancake with \(n\) cuts. - INPUT: - n– nonnegative integer
 - OUTPUT: integer; function value - EXAMPLES: - sage: a = sloane.A000124;a Central polygonal numbers (the Lazy Caterer's sequence): n(n+1)/2 + 1. sage: a(0) 1 sage: a(1) 2 sage: a(2) 4 sage: a(9) 46 sage: a.list(10) [1, 2, 4, 7, 11, 16, 22, 29, 37, 46] - >>> from sage.all import * >>> a = sloane.A000124;a Central polygonal numbers (the Lazy Caterer's sequence): n(n+1)/2 + 1. >>> a(Integer(0)) 1 >>> a(Integer(1)) 2 >>> a(Integer(2)) 4 >>> a(Integer(9)) 46 >>> a.list(Integer(10)) [1, 2, 4, 7, 11, 16, 22, 29, 37, 46] - AUTHORS: - Jaap Spies (2007-01-25) 
 
- class sage.combinat.sloane_functions.A000129[source]¶
- Bases: - RecurrenceSequence2- Pell numbers: \(a(0) = 0\), \(a(1) = 1\); for \(n > 1\), \(a(n) = 2a(n-1) + a(n-2)\). - Denominators of continued fraction convergents to \(\sqrt 2\). - See also A001333 - INPUT: - n– nonnegative integer
 - OUTPUT: integer; function value - EXAMPLES: - sage: a = sloane.A000129;a Pell numbers: a(0) = 0, a(1) = 1; for n > 1, a(n) = 2*a(n-1) + a(n-2). sage: a(0) 0 sage: a(2) 2 sage: a(12) 13860 sage: a.list(12) [0, 1, 2, 5, 12, 29, 70, 169, 408, 985, 2378, 5741] - >>> from sage.all import * >>> a = sloane.A000129;a Pell numbers: a(0) = 0, a(1) = 1; for n > 1, a(n) = 2*a(n-1) + a(n-2). >>> a(Integer(0)) 0 >>> a(Integer(2)) 2 >>> a(Integer(12)) 13860 >>> a.list(Integer(12)) [0, 1, 2, 5, 12, 29, 70, 169, 408, 985, 2378, 5741] - AUTHORS: - Jaap Spies (2007-01-25) 
 
- class sage.combinat.sloane_functions.A000142[source]¶
- Bases: - SloaneSequence- Factorial numbers: \(n! = 1 \cdot 2 \cdot 3 \cdots n\). - Order of symmetric group \(S_n\), number of permutations of \(n\) letters. - INPUT: - n– nonnegative integer
 - OUTPUT: integer; function value - EXAMPLES: - sage: a = sloane.A000142;a Factorial numbers: n! = 1*2*3*4*...*n (order of symmetric group S_n, number of permutations of n letters). sage: a(0) 1 sage: a(8) 40320 sage: a(40) 815915283247897734345611269596115894272000000000 sage: a.list(9) [1, 1, 2, 6, 24, 120, 720, 5040, 40320] - >>> from sage.all import * >>> a = sloane.A000142;a Factorial numbers: n! = 1*2*3*4*...*n (order of symmetric group S_n, number of permutations of n letters). >>> a(Integer(0)) 1 >>> a(Integer(8)) 40320 >>> a(Integer(40)) 815915283247897734345611269596115894272000000000 >>> a.list(Integer(9)) [1, 1, 2, 6, 24, 120, 720, 5040, 40320] - AUTHORS: - Jaap Spies (2007-01-12) 
 
- class sage.combinat.sloane_functions.A000153[source]¶
- Bases: - ExtremesOfPermanentsSequence- \(a(n) = n*a(n-1) + (n-2)*a(n-2)\), with \(a(0) = 0\), \(a(1) = 1\). - With offset 1, permanent of (0,1)-matrix of size \(n \times (n+d)\) with \(d=2\) and \(n\) zeros not on a line. This is a special case of Theorem 2.3 of Seok-Zun Song et al. Extremes of permanents of (0,1)-matrices, p. 201-202. - INPUT: - n– nonnegative integer
 - OUTPUT: integer; function value - EXAMPLES: - sage: a = sloane.A000153; a a(n) = n*a(n-1) + (n-2)*a(n-2), with a(0) = 0, a(1) = 1. sage: a(0) 0 sage: a(1) 1 sage: a(8) 82508 sage: a(20) 10315043624498196944 sage: a.list(8) [0, 1, 2, 7, 32, 181, 1214, 9403] - >>> from sage.all import * >>> a = sloane.A000153; a a(n) = n*a(n-1) + (n-2)*a(n-2), with a(0) = 0, a(1) = 1. >>> a(Integer(0)) 0 >>> a(Integer(1)) 1 >>> a(Integer(8)) 82508 >>> a(Integer(20)) 10315043624498196944 >>> a.list(Integer(8)) [0, 1, 2, 7, 32, 181, 1214, 9403] - AUTHORS: - Jaap Spies (2007-01-13) 
 
- class sage.combinat.sloane_functions.A000165[source]¶
- Bases: - SloaneSequence- Double factorial numbers: \((2n)!! = 2^n*n!\). - INPUT: - n– nonnegative integer
 - OUTPUT: integer; function value - EXAMPLES: - sage: a = sloane.A000165;a Double factorial numbers: (2n)!! = 2^n*n!. sage: a(0) 1 sage: a.offset 0 sage: a(8) 10321920 sage: a(20) 2551082656125828464640000 sage: a.list(9) [1, 2, 8, 48, 384, 3840, 46080, 645120, 10321920] - >>> from sage.all import * >>> a = sloane.A000165;a Double factorial numbers: (2n)!! = 2^n*n!. >>> a(Integer(0)) 1 >>> a.offset 0 >>> a(Integer(8)) 10321920 >>> a(Integer(20)) 2551082656125828464640000 >>> a.list(Integer(9)) [1, 2, 8, 48, 384, 3840, 46080, 645120, 10321920] - AUTHORS: - Jaap Spies (2007-01-24) 
 
- class sage.combinat.sloane_functions.A000166[source]¶
- Bases: - SloaneSequence- Subfactorial or rencontres numbers, or derangements: number of permutations of \(n\) elements with no fixed points. - With offset 1 also the permanent of a (0,1)-matrix of order \(n\) with \(n\) 0s not on a line. - INPUT: - n– nonnegative integer
 - OUTPUT: integer; function value - EXAMPLES: - sage: a = sloane.A000166;a Subfactorial or rencontres numbers, or derangements: number of permutations of $n$ elements with no fixed points. sage: a(0) 1 sage: a(1) 0 sage: a(2) 1 sage: a.offset 0 sage: a(8) 14833 sage: a(20) 895014631192902121 sage: a.list(9) [1, 0, 1, 2, 9, 44, 265, 1854, 14833] - >>> from sage.all import * >>> a = sloane.A000166;a Subfactorial or rencontres numbers, or derangements: number of permutations of $n$ elements with no fixed points. >>> a(Integer(0)) 1 >>> a(Integer(1)) 0 >>> a(Integer(2)) 1 >>> a.offset 0 >>> a(Integer(8)) 14833 >>> a(Integer(20)) 895014631192902121 >>> a.list(Integer(9)) [1, 0, 1, 2, 9, 44, 265, 1854, 14833] - AUTHORS: - Jaap Spies (2007-01-13) 
 
- class sage.combinat.sloane_functions.A000169[source]¶
- Bases: - SloaneSequence- Number of labeled rooted trees with \(n\) nodes: \(n^{(n-1)}\). - INPUT: - n– nonnegative integer
 - OUTPUT: integer; function value - EXAMPLES: - sage: a = sloane.A000169;a Number of labeled rooted trees with n nodes: n^(n-1). sage: a(0) Traceback (most recent call last): ... ValueError: input n (=0) must be a positive integer sage: a(1) 1 sage: a(2) 2 sage: a(10) 1000000000 sage: a.list(11) [1, 2, 9, 64, 625, 7776, 117649, 2097152, 43046721, 1000000000, 25937424601] - >>> from sage.all import * >>> a = sloane.A000169;a Number of labeled rooted trees with n nodes: n^(n-1). >>> a(Integer(0)) Traceback (most recent call last): ... ValueError: input n (=0) must be a positive integer >>> a(Integer(1)) 1 >>> a(Integer(2)) 2 >>> a(Integer(10)) 1000000000 >>> a.list(Integer(11)) [1, 2, 9, 64, 625, 7776, 117649, 2097152, 43046721, 1000000000, 25937424601] - AUTHORS: - Jaap Spies (2007-01-26) 
 
- class sage.combinat.sloane_functions.A000203[source]¶
- Bases: - SloaneSequence- The sequence \(\sigma(n)\), where \(\sigma(n)\) is the sum of the divisors of \(n\). Also called \(\sigma_1(n)\). - The function - sigma(n, k)implements \(\sigma_k(n)\) in Sage.- INPUT: - n– nonnegative integer
 - OUTPUT: integer; function value - EXAMPLES: - sage: a = sloane.A000203; a sigma(n) = sum of divisors of n. Also called sigma_1(n). sage: a(1) 1 sage: a(0) Traceback (most recent call last): ... ValueError: input n (=0) must be a positive integer sage: a(256) 511 sage: a.list(12) [1, 3, 4, 7, 6, 12, 8, 15, 13, 18, 12, 28] sage: a(1/3) Traceback (most recent call last): ... TypeError: input must be an int or Integer - >>> from sage.all import * >>> a = sloane.A000203; a sigma(n) = sum of divisors of n. Also called sigma_1(n). >>> a(Integer(1)) 1 >>> a(Integer(0)) Traceback (most recent call last): ... ValueError: input n (=0) must be a positive integer >>> a(Integer(256)) 511 >>> a.list(Integer(12)) [1, 3, 4, 7, 6, 12, 8, 15, 13, 18, 12, 28] >>> a(Integer(1)/Integer(3)) Traceback (most recent call last): ... TypeError: input must be an int or Integer - AUTHORS: - Jaap Spies (2007-01-13) 
 
- class sage.combinat.sloane_functions.A000204[source]¶
- Bases: - SloaneSequence- Lucas numbers (beginning with 1): \(L(n) = L(n-1) + L(n-2)\) with \(L(1) = 1\), \(L(2) = 3\). - INPUT: - n– nonnegative integer
 - OUTPUT: integer; function value - EXAMPLES: - sage: a = sloane.A000204; a Lucas numbers (beginning at 1): L(n) = L(n-1) + L(n-2), L(2) = 3. sage: a(1) 1 sage: a(8) 47 sage: a(200) 627376215338105766356982006981782561278127 sage: a(-4) Traceback (most recent call last): ... ValueError: input n (=-4) must be a positive integer sage: a.list(12) [1, 3, 4, 7, 11, 18, 29, 47, 76, 123, 199, 322] sage: a(0) Traceback (most recent call last): ... ValueError: input n (=0) must be a positive integer - >>> from sage.all import * >>> a = sloane.A000204; a Lucas numbers (beginning at 1): L(n) = L(n-1) + L(n-2), L(2) = 3. >>> a(Integer(1)) 1 >>> a(Integer(8)) 47 >>> a(Integer(200)) 627376215338105766356982006981782561278127 >>> a(-Integer(4)) Traceback (most recent call last): ... ValueError: input n (=-4) must be a positive integer >>> a.list(Integer(12)) [1, 3, 4, 7, 11, 18, 29, 47, 76, 123, 199, 322] >>> a(Integer(0)) Traceback (most recent call last): ... ValueError: input n (=0) must be a positive integer - AUTHORS: - Jaap Spies (2007-01-18) 
 
- class sage.combinat.sloane_functions.A000213[source]¶
- Bases: - SloaneSequence- Tribonacci numbers: a(n) = a(n-1) + a(n-2) + a(n-3). Starting with 1, 1, 1, … - INPUT: - n– nonnegative integer
 - OUTPUT: integer; function value - EXAMPLES: - sage: a = sloane.A000213;a Tribonacci numbers: a(n) = a(n-1) + a(n-2) + a(n-3). sage: a(0) 1 sage: a(1) 1 sage: a(2) 1 sage: a(11) 355 sage: a.list(12) [1, 1, 1, 3, 5, 9, 17, 31, 57, 105, 193, 355] - >>> from sage.all import * >>> a = sloane.A000213;a Tribonacci numbers: a(n) = a(n-1) + a(n-2) + a(n-3). >>> a(Integer(0)) 1 >>> a(Integer(1)) 1 >>> a(Integer(2)) 1 >>> a(Integer(11)) 355 >>> a.list(Integer(12)) [1, 1, 1, 3, 5, 9, 17, 31, 57, 105, 193, 355] - AUTHORS: - Jaap Spies (2007-01-19) 
 
- class sage.combinat.sloane_functions.A000217[source]¶
- Bases: - SloaneSequence- Triangular numbers: \(a(n) = \binom{n+1}{2} = n(n+1)/2\). - INPUT: - n– nonnegative integer
 - OUTPUT: integer; function value - EXAMPLES: - sage: a = sloane.A000217;a Triangular numbers: a(n) = C(n+1,2) = n(n+1)/2 = 0+1+2+...+n. sage: a(0) 0 sage: a(2) 3 sage: a(8) 36 sage: a(2000) 2001000 sage: a.list(9) [0, 1, 3, 6, 10, 15, 21, 28, 36] - >>> from sage.all import * >>> a = sloane.A000217;a Triangular numbers: a(n) = C(n+1,2) = n(n+1)/2 = 0+1+2+...+n. >>> a(Integer(0)) 0 >>> a(Integer(2)) 3 >>> a(Integer(8)) 36 >>> a(Integer(2000)) 2001000 >>> a.list(Integer(9)) [0, 1, 3, 6, 10, 15, 21, 28, 36] - AUTHORS: - Jaap Spies (2007-01-25) 
 
- class sage.combinat.sloane_functions.A000225[source]¶
- Bases: - SloaneSequence- \(2^n - 1\). - INPUT: - n– nonnegative integer
 - OUTPUT: integer; function value - EXAMPLES: - sage: a = sloane.A000225;a 2^n - 1. sage: a(0) 0 sage: a(-1) Traceback (most recent call last): ... ValueError: input n (=-1) must be an integer >= 0 sage: a(12) 4095 sage: a.list(12) [0, 1, 3, 7, 15, 31, 63, 127, 255, 511, 1023, 2047] - >>> from sage.all import * >>> a = sloane.A000225;a 2^n - 1. >>> a(Integer(0)) 0 >>> a(-Integer(1)) Traceback (most recent call last): ... ValueError: input n (=-1) must be an integer >= 0 >>> a(Integer(12)) 4095 >>> a.list(Integer(12)) [0, 1, 3, 7, 15, 31, 63, 127, 255, 511, 1023, 2047] - AUTHORS: - Jaap Spies (2007-01-25) 
 
- class sage.combinat.sloane_functions.A000244[source]¶
- Bases: - SloaneSequence- Powers of 3: \(a(n) = 3^n\). - INPUT: - n– nonnegative integer
 - OUTPUT: integer; function value - EXAMPLES: - sage: a = sloane.A000244;a Powers of 3: a(n) = 3^n. sage: a(-1) Traceback (most recent call last): ... ValueError: input n (=-1) must be an integer >= 0 sage: a(0) 1 sage: a(3) 27 sage: a(11) 177147 sage: a.list(12) [1, 3, 9, 27, 81, 243, 729, 2187, 6561, 19683, 59049, 177147] - >>> from sage.all import * >>> a = sloane.A000244;a Powers of 3: a(n) = 3^n. >>> a(-Integer(1)) Traceback (most recent call last): ... ValueError: input n (=-1) must be an integer >= 0 >>> a(Integer(0)) 1 >>> a(Integer(3)) 27 >>> a(Integer(11)) 177147 >>> a.list(Integer(12)) [1, 3, 9, 27, 81, 243, 729, 2187, 6561, 19683, 59049, 177147] - AUTHORS: - Jaap Spies (2007-01-26) 
 
- class sage.combinat.sloane_functions.A000255[source]¶
- Bases: - ExtremesOfPermanentsSequence- \(a(n) = n*a(n-1) + (n-1)*a(n-2)\), with \(a(0) = 1\), \(a(1) = 1\). - With offset 1, permanent of (0,1)-matrix of size \(n \times (n+d)\) with \(d=1\) and \(n\) zeros not on a line. This is a special case of Theorem 2.3 of Seok-Zun Song et al. Extremes of permanents of (0,1)-matrices, p. 201-202. - INPUT: - n– nonnegative integer
 - OUTPUT: integer; function value - EXAMPLES: - sage: a = sloane.A000255;a a(n) = n*a(n-1) + (n-1)*a(n-2), a(0) = 1, a(1) = 1. sage: a(0) 1 sage: a(1) 1 sage: a.offset 0 sage: a(8) 148329 sage: a(22) 9923922230666898717143 sage: a.list(9) [1, 1, 3, 11, 53, 309, 2119, 16687, 148329] - >>> from sage.all import * >>> a = sloane.A000255;a a(n) = n*a(n-1) + (n-1)*a(n-2), a(0) = 1, a(1) = 1. >>> a(Integer(0)) 1 >>> a(Integer(1)) 1 >>> a.offset 0 >>> a(Integer(8)) 148329 >>> a(Integer(22)) 9923922230666898717143 >>> a.list(Integer(9)) [1, 1, 3, 11, 53, 309, 2119, 16687, 148329] - AUTHORS: - Jaap Spies (2007-01-13) 
 
- class sage.combinat.sloane_functions.A000261[source]¶
- Bases: - ExtremesOfPermanentsSequence- \(a(n) = n*a(n-1) + (n-3)*a(n-2)\), with \(a(1) = 1\), \(a(2) = 1\). - With offset 1, permanent of (0,1)-matrix of size \(n \times (n+d)\) with \(d=3\) and \(n\) zeros not on a line. This is a special case of Theorem 2.3 of Seok-Zun Song et al. Extremes of permanents of (0,1)-matrices, p. 201-202. - Seok-Zun Song et al., Extremes of permanents of (0,1)-matrices, Lin. Algebra and its Applic. 373 (2003), p. 197-210. - INPUT: - n– nonnegative integer
 - OUTPUT: integer; function value - EXAMPLES: - sage: a = sloane.A000261;a a(n) = n*a(n-1) + (n-3)*a(n-2), a(1) = 0, a(2) = 1. sage: a(0) Traceback (most recent call last): ... ValueError: input n (=0) must be a positive integer sage: a(1) 0 sage: a.offset 1 sage: a(8) 30637 sage: a(22) 1801366114380914335441 sage: a.list(9) [0, 1, 3, 13, 71, 465, 3539, 30637, 296967] - >>> from sage.all import * >>> a = sloane.A000261;a a(n) = n*a(n-1) + (n-3)*a(n-2), a(1) = 0, a(2) = 1. >>> a(Integer(0)) Traceback (most recent call last): ... ValueError: input n (=0) must be a positive integer >>> a(Integer(1)) 0 >>> a.offset 1 >>> a(Integer(8)) 30637 >>> a(Integer(22)) 1801366114380914335441 >>> a.list(Integer(9)) [0, 1, 3, 13, 71, 465, 3539, 30637, 296967] - AUTHORS: - Jaap Spies (2007-01-23) 
 
- class sage.combinat.sloane_functions.A000272[source]¶
- Bases: - SloaneSequence- Number of labeled rooted trees on \(n\) nodes: \(n^{(n-2)}\). - INPUT: - n– integer
 - OUTPUT: integer; function value - EXAMPLES: - sage: a = sloane.A000272;a Number of labeled rooted trees with n nodes: n^(n-2). sage: a(0) 1 sage: a(1) 1 sage: a(2) 1 sage: a(10) 100000000 sage: a.list(12) [1, 1, 1, 3, 16, 125, 1296, 16807, 262144, 4782969, 100000000, 2357947691] - >>> from sage.all import * >>> a = sloane.A000272;a Number of labeled rooted trees with n nodes: n^(n-2). >>> a(Integer(0)) 1 >>> a(Integer(1)) 1 >>> a(Integer(2)) 1 >>> a(Integer(10)) 100000000 >>> a.list(Integer(12)) [1, 1, 1, 3, 16, 125, 1296, 16807, 262144, 4782969, 100000000, 2357947691] - AUTHORS: - Jaap Spies (2007-01-26) 
 
- class sage.combinat.sloane_functions.A000290[source]¶
- Bases: - SloaneSequence- The squares: \(a(n) = n^2\). - INPUT: - n– nonnegative integer
 - OUTPUT: integer; function value - EXAMPLES: - sage: a = sloane.A000290;a The squares: a(n) = n^2. sage: a(0) 0 sage: a(-1) Traceback (most recent call last): ... ValueError: input n (=-1) must be an integer >= 0 sage: a(16) 256 sage: a.list(17) [0, 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225, 256] - >>> from sage.all import * >>> a = sloane.A000290;a The squares: a(n) = n^2. >>> a(Integer(0)) 0 >>> a(-Integer(1)) Traceback (most recent call last): ... ValueError: input n (=-1) must be an integer >= 0 >>> a(Integer(16)) 256 >>> a.list(Integer(17)) [0, 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225, 256] - AUTHORS: - Jaap Spies (2007-01-25) 
 
- class sage.combinat.sloane_functions.A000292[source]¶
- Bases: - SloaneSequence- Tetrahedral (or pyramidal) numbers: \(\binom{n+2}{3} = n(n+1)(n+2)/6\). - INPUT: - n– nonnegative integer
 - OUTPUT: integer; function value - EXAMPLES: - sage: a = sloane.A000292;a Tetrahedral (or pyramidal) numbers: C(n+2,3) = n(n+1)(n+2)/6. sage: a(0) 0 sage: a(2) 4 sage: a(11) 286 sage: a.list(12) [0, 1, 4, 10, 20, 35, 56, 84, 120, 165, 220, 286] - >>> from sage.all import * >>> a = sloane.A000292;a Tetrahedral (or pyramidal) numbers: C(n+2,3) = n(n+1)(n+2)/6. >>> a(Integer(0)) 0 >>> a(Integer(2)) 4 >>> a(Integer(11)) 286 >>> a.list(Integer(12)) [0, 1, 4, 10, 20, 35, 56, 84, 120, 165, 220, 286] - AUTHORS: - Jaap Spies (2007-01-26) 
 
- class sage.combinat.sloane_functions.A000302[source]¶
- Bases: - SloaneSequence- Powers of 4: \(a(n) = 4^n\). - INPUT: - n– nonnegative integer
 - OUTPUT: integer; function value - EXAMPLES: - sage: a = sloane.A000302;a Powers of 4: a(n) = 4^n. sage: a(0) 1 sage: a(1) 4 sage: a(2) 16 sage: a(10) 1048576 sage: a.list(12) [1, 4, 16, 64, 256, 1024, 4096, 16384, 65536, 262144, 1048576, 4194304] - >>> from sage.all import * >>> a = sloane.A000302;a Powers of 4: a(n) = 4^n. >>> a(Integer(0)) 1 >>> a(Integer(1)) 4 >>> a(Integer(2)) 16 >>> a(Integer(10)) 1048576 >>> a.list(Integer(12)) [1, 4, 16, 64, 256, 1024, 4096, 16384, 65536, 262144, 1048576, 4194304] - AUTHORS: - Jaap Spies (2007-01-26) 
 
- class sage.combinat.sloane_functions.A000312[source]¶
- Bases: - SloaneSequence- Number of labeled mappings from \(n\) points to themselves (endofunctions): \(n^n\). - INPUT: - n– nonnegative integer
 - OUTPUT: integer; function value - EXAMPLES: - sage: a = sloane.A000312;a Number of labeled mappings from n points to themselves (endofunctions): n^n. sage: a(-1) Traceback (most recent call last): ... ValueError: input n (=-1) must be an integer >= 0 sage: a(0) 1 sage: a(1) 1 sage: a(9) 387420489 sage: a.list(11) [1, 1, 4, 27, 256, 3125, 46656, 823543, 16777216, 387420489, 10000000000] - >>> from sage.all import * >>> a = sloane.A000312;a Number of labeled mappings from n points to themselves (endofunctions): n^n. >>> a(-Integer(1)) Traceback (most recent call last): ... ValueError: input n (=-1) must be an integer >= 0 >>> a(Integer(0)) 1 >>> a(Integer(1)) 1 >>> a(Integer(9)) 387420489 >>> a.list(Integer(11)) [1, 1, 4, 27, 256, 3125, 46656, 823543, 16777216, 387420489, 10000000000] - AUTHORS: - Jaap Spies (2007-01-26) 
 
- class sage.combinat.sloane_functions.A000326[source]¶
- Bases: - SloaneSequence- Pentagonal numbers: \(n(3n-1)/2\). - INPUT: - n– nonnegative integer
 - OUTPUT: integer; function value - EXAMPLES: - sage: a = sloane.A000326;a Pentagonal numbers: n(3n-1)/2. sage: a(0) 0 sage: a(1) 1 sage: a(2) 5 sage: a(10) 145 sage: a.list(12) [0, 1, 5, 12, 22, 35, 51, 70, 92, 117, 145, 176] sage: a(1/3) Traceback (most recent call last): ... TypeError: input must be an int or Integer - >>> from sage.all import * >>> a = sloane.A000326;a Pentagonal numbers: n(3n-1)/2. >>> a(Integer(0)) 0 >>> a(Integer(1)) 1 >>> a(Integer(2)) 5 >>> a(Integer(10)) 145 >>> a.list(Integer(12)) [0, 1, 5, 12, 22, 35, 51, 70, 92, 117, 145, 176] >>> a(Integer(1)/Integer(3)) Traceback (most recent call last): ... TypeError: input must be an int or Integer - AUTHORS: - Jaap Spies (2007-01-26) 
 
- class sage.combinat.sloane_functions.A000330[source]¶
- Bases: - SloaneSequence- Square pyramidal numbers” \(0^2 + 1^2 \cdots n^2 = n(n+1)(2n+1)/6\). - INPUT: - n– nonnegative integer
 - OUTPUT: integer; function value - EXAMPLES: - sage: a = sloane.A000330;a Square pyramidal numbers: 0^2+1^2+2^2+...+n^2 = n(n+1)(2n+1)/6. sage: a(-1) Traceback (most recent call last): ... ValueError: input n (=-1) must be an integer >= 0 sage: a(0) 0 sage: a(3) 14 sage: a(11) 506 sage: a.list(12) [0, 1, 5, 14, 30, 55, 91, 140, 204, 285, 385, 506] - >>> from sage.all import * >>> a = sloane.A000330;a Square pyramidal numbers: 0^2+1^2+2^2+...+n^2 = n(n+1)(2n+1)/6. >>> a(-Integer(1)) Traceback (most recent call last): ... ValueError: input n (=-1) must be an integer >= 0 >>> a(Integer(0)) 0 >>> a(Integer(3)) 14 >>> a(Integer(11)) 506 >>> a.list(Integer(12)) [0, 1, 5, 14, 30, 55, 91, 140, 204, 285, 385, 506] - AUTHORS: - Jaap Spies (2007-01-26) 
 
- class sage.combinat.sloane_functions.A000396[source]¶
- Bases: - SloaneSequence- Perfect numbers: equal to sum of proper divisors. - INPUT: - n– nonnegative integer
 - OUTPUT: integer; function value - EXAMPLES: - sage: a = sloane.A000396;a Perfect numbers: equal to sum of proper divisors. sage: a(0) Traceback (most recent call last): ... ValueError: input n (=0) must be a positive integer sage: a(1) 6 sage: a(2) 28 sage: a(7) 137438691328 sage: a.list(7) [6, 28, 496, 8128, 33550336, 8589869056, 137438691328] - >>> from sage.all import * >>> a = sloane.A000396;a Perfect numbers: equal to sum of proper divisors. >>> a(Integer(0)) Traceback (most recent call last): ... ValueError: input n (=0) must be a positive integer >>> a(Integer(1)) 6 >>> a(Integer(2)) 28 >>> a(Integer(7)) 137438691328 >>> a.list(Integer(7)) [6, 28, 496, 8128, 33550336, 8589869056, 137438691328] - AUTHORS: - Jaap Spies (2007-01-25) 
 
- class sage.combinat.sloane_functions.A000578[source]¶
- Bases: - SloaneSequence- The cubes: \(a(n) = n^3\). - INPUT: - n– nonnegative integer
 - OUTPUT: integer; function value - EXAMPLES: - sage: a = sloane.A000578;a The cubes: n^3 sage: a(-1) Traceback (most recent call last): ... ValueError: input n (=-1) must be an integer >= 0 sage: a(0) 0 sage: a(3) 27 sage: a(11) 1331 sage: a.list(12) [0, 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000, 1331] - >>> from sage.all import * >>> a = sloane.A000578;a The cubes: n^3 >>> a(-Integer(1)) Traceback (most recent call last): ... ValueError: input n (=-1) must be an integer >= 0 >>> a(Integer(0)) 0 >>> a(Integer(3)) 27 >>> a(Integer(11)) 1331 >>> a.list(Integer(12)) [0, 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000, 1331] - AUTHORS: - Jaap Spies (2007-01-26) 
 
- class sage.combinat.sloane_functions.A000583[source]¶
- Bases: - SloaneSequence- Fourth powers: \(a(n) = n^4\). - INPUT: - n– nonnegative integer
 - OUTPUT: integer; function value - EXAMPLES: - sage: a = sloane.A000583;a Fourth powers: n^4. sage: a(0.0) Traceback (most recent call last): ... TypeError: input must be an int or Integer sage: a(1) 1 sage: a(2) 16 sage: a(9) 6561 sage: a.list(10) [0, 1, 16, 81, 256, 625, 1296, 2401, 4096, 6561] - >>> from sage.all import * >>> a = sloane.A000583;a Fourth powers: n^4. >>> a(RealNumber('0.0')) Traceback (most recent call last): ... TypeError: input must be an int or Integer >>> a(Integer(1)) 1 >>> a(Integer(2)) 16 >>> a(Integer(9)) 6561 >>> a.list(Integer(10)) [0, 1, 16, 81, 256, 625, 1296, 2401, 4096, 6561] - AUTHORS: - Jaap Spies (2007-02-04) 
 
- class sage.combinat.sloane_functions.A000587[source]¶
- Bases: - ExponentialNumbers- The sequence of Uppuluri-Carpenter numbers. - The Uppuluri-Carpenter number \(C_n\) counts the imbalance in the number of ways to put \(n\) distinguishable things into an even number of indistinguishable boxes versus into an odd number of indistinguishable boxes, such that no box is empty. - Let \(S(n, k)\) denote the Stirling number of the second kind. Then \[C_n = \sum{k=0}^{n} (-1)^k S(n, k) .\]- INPUT: - n– nonnegative integer
 - OUTPUT: integer; \(C_n\) - EXAMPLES: - sage: a = sloane.A000587; a Sequence of Uppuluri-Carpenter numbers sage: a.offset 0 sage: a(0) 1 sage: a(100) 397577026456518507969762382254187048845620355238545130875069912944235105204434466095862371032124545552161 sage: a.list(10) [1, -1, 0, 1, 1, -2, -9, -9, 50, 267] - >>> from sage.all import * >>> a = sloane.A000587; a Sequence of Uppuluri-Carpenter numbers >>> a.offset 0 >>> a(Integer(0)) 1 >>> a(Integer(100)) 397577026456518507969762382254187048845620355238545130875069912944235105204434466095862371032124545552161 >>> a.list(Integer(10)) [1, -1, 0, 1, 1, -2, -9, -9, 50, 267] - AUTHORS: - Nick Alexander 
 
- class sage.combinat.sloane_functions.A000668[source]¶
- Bases: - SloaneSequence- Mersenne primes (of form \(2^p - 1\) where \(p\) is a prime). - (See A000043 for the values of \(p\).) - Warning: a(39) has 4,053,946 digits! - INPUT: - n– nonnegative integer
 - OUTPUT: integer; function value - EXAMPLES: - sage: a = sloane.A000668;a Mersenne primes (of form 2^p - 1 where p is a prime). (See A000043 for the values of p.) sage: a(1) 3 sage: a(2) 7 sage: a(12) 170141183460469231731687303715884105727 - >>> from sage.all import * >>> a = sloane.A000668;a Mersenne primes (of form 2^p - 1 where p is a prime). (See A000043 for the values of p.) >>> a(Integer(1)) 3 >>> a(Integer(2)) 7 >>> a(Integer(12)) 170141183460469231731687303715884105727 - Warning: a(39) has 4,053,946 digits! - sage: a(40) Traceback (most recent call last): ... IndexError: list index out of range sage: a.list(8) [3, 7, 31, 127, 8191, 131071, 524287, 2147483647] - >>> from sage.all import * >>> a(Integer(40)) Traceback (most recent call last): ... IndexError: list index out of range >>> a.list(Integer(8)) [3, 7, 31, 127, 8191, 131071, 524287, 2147483647] - AUTHORS: - Jaap Spies (2007-01-25) 
 
- class sage.combinat.sloane_functions.A000670[source]¶
- Bases: - SloaneSequence- Number of preferential arrangements of \(n\) labeled elements; or number of weak orders on \(n\) labeled elements. - INPUT: - n– nonnegative integer
 - OUTPUT: integer; function value - EXAMPLES: - sage: a = sloane.A000670;a Number of preferential arrangements of n labeled elements. sage: a(0) 1 sage: a(1) 1 sage: a(2) 3 sage: a(9) 7087261 sage: a.list(10) [1, 1, 3, 13, 75, 541, 4683, 47293, 545835, 7087261] - >>> from sage.all import * >>> a = sloane.A000670;a Number of preferential arrangements of n labeled elements. >>> a(Integer(0)) 1 >>> a(Integer(1)) 1 >>> a(Integer(2)) 3 >>> a(Integer(9)) 7087261 >>> a.list(Integer(10)) [1, 1, 3, 13, 75, 541, 4683, 47293, 545835, 7087261] - AUTHORS: - Jaap Spies (2007-02-03) 
 
- class sage.combinat.sloane_functions.A000720[source]¶
- Bases: - SloaneSequence- \(pi(n)\), the number of primes \(\le n\). Sometimes called \(PrimePi(n)\). - INPUT: - n– nonnegative integer
 - OUTPUT: integer; function value - EXAMPLES: - sage: a = sloane.A000720;a pi(n), the number of primes <= n. Sometimes called PrimePi(n) sage: a(0) Traceback (most recent call last): ... ValueError: input n (=0) must be a positive integer sage: a(2) 1 sage: a(8) 4 sage: a(1000) 168 sage: a.list(12) [0, 1, 2, 2, 3, 3, 4, 4, 4, 4, 5, 5] - >>> from sage.all import * >>> a = sloane.A000720;a pi(n), the number of primes <= n. Sometimes called PrimePi(n) >>> a(Integer(0)) Traceback (most recent call last): ... ValueError: input n (=0) must be a positive integer >>> a(Integer(2)) 1 >>> a(Integer(8)) 4 >>> a(Integer(1000)) 168 >>> a.list(Integer(12)) [0, 1, 2, 2, 3, 3, 4, 4, 4, 4, 5, 5] - AUTHORS: - Jaap Spies (2007-01-25) 
 
- class sage.combinat.sloane_functions.A000796[source]¶
- Bases: - SloaneSequence- Decimal expansion of \(\pi\). - INPUT: - n– positive integer
 - OUTPUT: integer; function value - EXAMPLES: - sage: a = sloane.A000796;a Decimal expansion of Pi. sage: a(0) Traceback (most recent call last): ... ValueError: input n (=0) must be a positive integer sage: a(1) 3 sage: a(13) 9 sage: a.list(14) [3, 1, 4, 1, 5, 9, 2, 6, 5, 3, 5, 8, 9, 7] sage: a(100) 7 - >>> from sage.all import * >>> a = sloane.A000796;a Decimal expansion of Pi. >>> a(Integer(0)) Traceback (most recent call last): ... ValueError: input n (=0) must be a positive integer >>> a(Integer(1)) 3 >>> a(Integer(13)) 9 >>> a.list(Integer(14)) [3, 1, 4, 1, 5, 9, 2, 6, 5, 3, 5, 8, 9, 7] >>> a(Integer(100)) 7 - AUTHOR: - Jaap Spies (2007-01-30) 
 - list(n)[source]¶
- EXAMPLES: - sage: sloane.A000796.list(10) [3, 1, 4, 1, 5, 9, 2, 6, 5, 3] - >>> from sage.all import * >>> sloane.A000796.list(Integer(10)) [3, 1, 4, 1, 5, 9, 2, 6, 5, 3] 
 - pi()[source]¶
- Based on an algorithm of Lambert Meertens The ABC-programming language!!! - EXAMPLES: - sage: it = sloane.A000796.pi() sage: [next(it) for i in range(10)] [3, 1, 4, 1, 5, 9, 2, 6, 5, 3] - >>> from sage.all import * >>> it = sloane.A000796.pi() >>> [next(it) for i in range(Integer(10))] [3, 1, 4, 1, 5, 9, 2, 6, 5, 3] 
 
- class sage.combinat.sloane_functions.A000961[source]¶
- Bases: - SloaneSequence- Prime powers. - INPUT: - n– nonnegative integer
 - OUTPUT: integer; function value - EXAMPLES: - sage: a = sloane.A000961;a Prime powers. sage: a(0) Traceback (most recent call last): ... ValueError: input n (=0) must be a positive integer sage: a(2) 2 sage: a(12) 17 sage: a.list(12) [1, 2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17] - >>> from sage.all import * >>> a = sloane.A000961;a Prime powers. >>> a(Integer(0)) Traceback (most recent call last): ... ValueError: input n (=0) must be a positive integer >>> a(Integer(2)) 2 >>> a(Integer(12)) 17 >>> a.list(Integer(12)) [1, 2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17] - AUTHORS: - Jaap Spies (2007-01-25) 
 
- class sage.combinat.sloane_functions.A000984[source]¶
- Bases: - SloaneSequence- Central binomial coefficients: \(\binom{2n}{n} = \frac {(2n)!} {(n!)^2}\). - INPUT: - n– nonnegative integer
 - OUTPUT: integer; function value - EXAMPLES: - sage: a = sloane.A000984;a Central binomial coefficients: C(2n,n) = (2n)!/(n!)^2 sage: a(0) 1 sage: a(2) 6 sage: a(8) 12870 sage: a.list(9) [1, 2, 6, 20, 70, 252, 924, 3432, 12870] - >>> from sage.all import * >>> a = sloane.A000984;a Central binomial coefficients: C(2n,n) = (2n)!/(n!)^2 >>> a(Integer(0)) 1 >>> a(Integer(2)) 6 >>> a(Integer(8)) 12870 >>> a.list(Integer(9)) [1, 2, 6, 20, 70, 252, 924, 3432, 12870] - AUTHORS: - Jaap Spies (2007-01-26) 
 
- class sage.combinat.sloane_functions.A001006[source]¶
- Bases: - SloaneSequence- Motzkin numbers: number of ways of drawing any number of nonintersecting chords among \(n\) points on a circle. - INPUT: - n– nonnegative integer
 - OUTPUT: integer; function value - EXAMPLES: - sage: a = sloane.A001006;a Motzkin numbers: number of ways of drawing any number of nonintersecting chords among n points on a circle. sage: a(0) 1 sage: a(1) 1 sage: a(2) 2 sage: a(12) 15511 sage: a.list(13) [1, 1, 2, 4, 9, 21, 51, 127, 323, 835, 2188, 5798, 15511] - >>> from sage.all import * >>> a = sloane.A001006;a Motzkin numbers: number of ways of drawing any number of nonintersecting chords among n points on a circle. >>> a(Integer(0)) 1 >>> a(Integer(1)) 1 >>> a(Integer(2)) 2 >>> a(Integer(12)) 15511 >>> a.list(Integer(13)) [1, 1, 2, 4, 9, 21, 51, 127, 323, 835, 2188, 5798, 15511] - AUTHORS: - Jaap Spies (2007-02-02) 
 
- class sage.combinat.sloane_functions.A001045[source]¶
- Bases: - RecurrenceSequence2- Jacobsthal sequence: \(a(n) = a(n-1) + 2a(n-2)\), \(a(0) = 0\) and \(a(1) = 1\). - INPUT: - n– nonnegative integer
 - OUTPUT: integer; function value - EXAMPLES: - sage: a = sloane.A001045;a Jacobsthal sequence: a(n) = a(n-1) + 2a(n-2). sage: a(0) 0 sage: a(1) 1 sage: a(2) 1 sage: a(11) 683 sage: a.list(12) [0, 1, 1, 3, 5, 11, 21, 43, 85, 171, 341, 683] - >>> from sage.all import * >>> a = sloane.A001045;a Jacobsthal sequence: a(n) = a(n-1) + 2a(n-2). >>> a(Integer(0)) 0 >>> a(Integer(1)) 1 >>> a(Integer(2)) 1 >>> a(Integer(11)) 683 >>> a.list(Integer(12)) [0, 1, 1, 3, 5, 11, 21, 43, 85, 171, 341, 683] - AUTHORS: - Jaap Spies (2007-01-26) 
 
- class sage.combinat.sloane_functions.A001055[source]¶
- Bases: - SloaneSequence- Number of ways of factoring \(n\) with all factors 1. - INPUT: - n– nonnegative integer
 - OUTPUT: integer; function value - EXAMPLES: - sage: a = sloane.A001055;a Number of ways of factoring n with all factors >1. sage: a(0) Traceback (most recent call last): ... ValueError: input n (=0) must be a positive integer sage: a(1) 1 sage: a(2) 1 sage: a(9) 2 sage: a.list(16) [1, 1, 1, 2, 1, 2, 1, 3, 2, 2, 1, 4, 1, 2, 2, 5] - >>> from sage.all import * >>> a = sloane.A001055;a Number of ways of factoring n with all factors >1. >>> a(Integer(0)) Traceback (most recent call last): ... ValueError: input n (=0) must be a positive integer >>> a(Integer(1)) 1 >>> a(Integer(2)) 1 >>> a(Integer(9)) 2 >>> a.list(Integer(16)) [1, 1, 1, 2, 1, 2, 1, 3, 2, 2, 1, 4, 1, 2, 2, 5] - AUTHORS: - Jaap Spies (2007-02-04) 
 - nwf(n, m)[source]¶
- EXAMPLES: - sage: sloane.A001055.nwf(4,1) 0 sage: sloane.A001055.nwf(4,2) 1 sage: sloane.A001055.nwf(4,3) 1 sage: sloane.A001055.nwf(4,4) 2 - >>> from sage.all import * >>> sloane.A001055.nwf(Integer(4),Integer(1)) 0 >>> sloane.A001055.nwf(Integer(4),Integer(2)) 1 >>> sloane.A001055.nwf(Integer(4),Integer(3)) 1 >>> sloane.A001055.nwf(Integer(4),Integer(4)) 2 
 
- class sage.combinat.sloane_functions.A001109[source]¶
- Bases: - RecurrenceSequence2- \(a(n)^2\) is a triangular number: \(a(n) = 6*a(n-1) - a(n-2)\) with \(a(0)=0\), \(a(1)=1\). - INPUT: - n– nonnegative integer
 - OUTPUT: integer; function value - EXAMPLES: - sage: a = sloane.A001109;a a(n)^2 is a triangular number: a(n) = 6*a(n-1) - a(n-2) with a(0)=0, a(1)=1 sage: a(0) 0 sage: a(1) 1 sage: a(2) 6 sage: a.offset 0 sage: a(8) 235416 sage: a(60) 1515330104844857898115857393785728383101709300 sage: a.list(9) [0, 1, 6, 35, 204, 1189, 6930, 40391, 235416] - >>> from sage.all import * >>> a = sloane.A001109;a a(n)^2 is a triangular number: a(n) = 6*a(n-1) - a(n-2) with a(0)=0, a(1)=1 >>> a(Integer(0)) 0 >>> a(Integer(1)) 1 >>> a(Integer(2)) 6 >>> a.offset 0 >>> a(Integer(8)) 235416 >>> a(Integer(60)) 1515330104844857898115857393785728383101709300 >>> a.list(Integer(9)) [0, 1, 6, 35, 204, 1189, 6930, 40391, 235416] - AUTHORS: - Jaap Spies (2007-01-24) 
 
- class sage.combinat.sloane_functions.A001110[source]¶
- Bases: - RecurrenceSequence- Numbers that are both triangular and square: \(a(n) = 34a(n-1) - a(n-2) + 2\). - INPUT: - n– nonnegative integer
 - OUTPUT: integer; function value - EXAMPLES: - sage: a = sloane.A001110; a Numbers that are both triangular and square: a(n) = 34a(n-1) - a(n-2) + 2. sage: a(0) 0 sage: a(1) 1 sage: a(8) 55420693056 sage: a(21) 4446390382511295358038307980025 sage: a.list(8) [0, 1, 36, 1225, 41616, 1413721, 48024900, 1631432881] - >>> from sage.all import * >>> a = sloane.A001110; a Numbers that are both triangular and square: a(n) = 34a(n-1) - a(n-2) + 2. >>> a(Integer(0)) 0 >>> a(Integer(1)) 1 >>> a(Integer(8)) 55420693056 >>> a(Integer(21)) 4446390382511295358038307980025 >>> a.list(Integer(8)) [0, 1, 36, 1225, 41616, 1413721, 48024900, 1631432881] - AUTHORS: - Jaap Spies (2007-01-19) 
 - g(k)[source]¶
- EXAMPLES: - sage: sloane.A001110.g(2) 2 sage: sloane.A001110.g(1) 0 - >>> from sage.all import * >>> sloane.A001110.g(Integer(2)) 2 >>> sloane.A001110.g(Integer(1)) 0 
 - link = 'http://oeis.org/classic/A001110'¶
 
- class sage.combinat.sloane_functions.A001147[source]¶
- Bases: - SloaneSequence- Double factorial numbers: \((2n-1)!! = 1 \cdot 3 \cdot 5 \cdots (2n-1)\). - INPUT: - n– nonnegative integer
 - OUTPUT: integer; function value - EXAMPLES: - sage: a = sloane.A001147;a Double factorial numbers: (2n-1)!! = 1.3.5....(2n-1). sage: a(0) 1 sage: a.offset 0 sage: a(8) 2027025 sage: a(20) 319830986772877770815625 sage: a.list(9) [1, 1, 3, 15, 105, 945, 10395, 135135, 2027025] - >>> from sage.all import * >>> a = sloane.A001147;a Double factorial numbers: (2n-1)!! = 1.3.5....(2n-1). >>> a(Integer(0)) 1 >>> a.offset 0 >>> a(Integer(8)) 2027025 >>> a(Integer(20)) 319830986772877770815625 >>> a.list(Integer(9)) [1, 1, 3, 15, 105, 945, 10395, 135135, 2027025] - AUTHORS: - Jaap Spies (2007-01-24) 
 
- class sage.combinat.sloane_functions.A001157[source]¶
- Bases: - SloaneSequence- The sequence \(\sigma_2(n)\), sum of squares of divisors of \(n\). - The function sigma(n, k) implements \(\sigma_k*\) in Sage. - INPUT: - n– nonnegative integer
 - OUTPUT: integer; function value - EXAMPLES: - sage: a = sloane.A001157;a sigma_2(n): sum of squares of divisors of n sage: a(0) Traceback (most recent call last): ... ValueError: input n (=0) must be a positive integer sage: a(2) 5 sage: a(8) 85 sage: a.list(9) [1, 5, 10, 21, 26, 50, 50, 85, 91] - >>> from sage.all import * >>> a = sloane.A001157;a sigma_2(n): sum of squares of divisors of n >>> a(Integer(0)) Traceback (most recent call last): ... ValueError: input n (=0) must be a positive integer >>> a(Integer(2)) 5 >>> a(Integer(8)) 85 >>> a.list(Integer(9)) [1, 5, 10, 21, 26, 50, 50, 85, 91] - AUTHORS: - Jaap Spies (2007-01-13) 
 
- class sage.combinat.sloane_functions.A001189[source]¶
- Bases: - SloaneSequence- Number of degree-n permutations of order exactly 2. - INPUT: - n– nonnegative integer
 - OUTPUT: integer; function value - EXAMPLES: - sage: a = sloane.A001189;a Number of degree-n permutations of order exactly 2. sage: a(0) Traceback (most recent call last): ... ValueError: input n (=0) must be a positive integer sage: a(1) 0 sage: a(2) 1 sage: a(12) 140151 sage: a.list(13) [0, 1, 3, 9, 25, 75, 231, 763, 2619, 9495, 35695, 140151, 568503] - >>> from sage.all import * >>> a = sloane.A001189;a Number of degree-n permutations of order exactly 2. >>> a(Integer(0)) Traceback (most recent call last): ... ValueError: input n (=0) must be a positive integer >>> a(Integer(1)) 0 >>> a(Integer(2)) 1 >>> a(Integer(12)) 140151 >>> a.list(Integer(13)) [0, 1, 3, 9, 25, 75, 231, 763, 2619, 9495, 35695, 140151, 568503] - AUTHORS: - Jaap Spies (2007-02-03) 
 
- class sage.combinat.sloane_functions.A001221[source]¶
- Bases: - SloaneSequence- Number of different prime divisors of \(n\). - Also called omega(n) or \(\omega(n)\). Maximal number of terms in any factorization of \(n\). Number of prime powers that divide \(n\). - INPUT: - n– nonnegative integer
 - OUTPUT: integer; function value - EXAMPLES: - sage: a = sloane.A001221; a Number of distinct primes dividing n (also called omega(n)). sage: a(0) Traceback (most recent call last): ... ValueError: input n (=0) must be a positive integer sage: a(1) 0 sage: a(8) 1 sage: a(41) 1 sage: a(84792) 3 sage: a.list(12) [0, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2] - >>> from sage.all import * >>> a = sloane.A001221; a Number of distinct primes dividing n (also called omega(n)). >>> a(Integer(0)) Traceback (most recent call last): ... ValueError: input n (=0) must be a positive integer >>> a(Integer(1)) 0 >>> a(Integer(8)) 1 >>> a(Integer(41)) 1 >>> a(Integer(84792)) 3 >>> a.list(Integer(12)) [0, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2] - AUTHORS: - Jaap Spies (2007-01-19) 
 
- class sage.combinat.sloane_functions.A001222[source]¶
- Bases: - SloaneSequence- Number of prime divisors of \(n\) (counted with multiplicity). - Also called bigomega(n) or \(\Omega(n)\). Maximal number of terms in any factorization of \(n\). Number of prime powers that divide \(n\). - INPUT: - n– nonnegative integer
 - OUTPUT: integer; function value - EXAMPLES: - sage: a = sloane.A001222; a Number of prime divisors of n (counted with multiplicity). sage: a(0) Traceback (most recent call last): ... ValueError: input n (=0) must be a positive integer sage: a(1) 0 sage: a(8) 3 sage: a(41) 1 sage: a(84792) 5 sage: a.list(12) [0, 1, 1, 2, 1, 2, 1, 3, 2, 2, 1, 3] - >>> from sage.all import * >>> a = sloane.A001222; a Number of prime divisors of n (counted with multiplicity). >>> a(Integer(0)) Traceback (most recent call last): ... ValueError: input n (=0) must be a positive integer >>> a(Integer(1)) 0 >>> a(Integer(8)) 3 >>> a(Integer(41)) 1 >>> a(Integer(84792)) 5 >>> a.list(Integer(12)) [0, 1, 1, 2, 1, 2, 1, 3, 2, 2, 1, 3] - AUTHORS: - Jaap Spies (2007-01-19) 
 
- class sage.combinat.sloane_functions.A001227[source]¶
- Bases: - SloaneSequence- Number of odd divisors of \(n\). - INPUT: - n– nonnegative integer
 - OUTPUT: integer; function value - EXAMPLES: - sage: a = sloane.A001227; a Number of odd divisors of n sage: a.offset 1 sage: a(1) 1 sage: a(0) Traceback (most recent call last): ... ValueError: input n (=0) must be a positive integer sage: a(100) 3 sage: a(256) 1 sage: a(29) 2 sage: a.list(20) [1, 1, 2, 1, 2, 2, 2, 1, 3, 2, 2, 2, 2, 2, 4, 1, 2, 3, 2, 2] sage: a(-1) Traceback (most recent call last): ... ValueError: input n (=-1) must be a positive integer - >>> from sage.all import * >>> a = sloane.A001227; a Number of odd divisors of n >>> a.offset 1 >>> a(Integer(1)) 1 >>> a(Integer(0)) Traceback (most recent call last): ... ValueError: input n (=0) must be a positive integer >>> a(Integer(100)) 3 >>> a(Integer(256)) 1 >>> a(Integer(29)) 2 >>> a.list(Integer(20)) [1, 1, 2, 1, 2, 2, 2, 1, 3, 2, 2, 2, 2, 2, 4, 1, 2, 3, 2, 2] >>> a(-Integer(1)) Traceback (most recent call last): ... ValueError: input n (=-1) must be a positive integer - AUTHORS: - Jaap Spies (2007-01-14) 
 
- class sage.combinat.sloane_functions.A001333[source]¶
- Bases: - RecurrenceSequence2- Numerators of continued fraction convergents to \(\sqrt 2\). - See also A000129 - INPUT: - n– nonnegative integer
 - OUTPUT: integer; function value - EXAMPLES: - sage: a = sloane.A001333;a Numerators of continued fraction convergents to sqrt(2). sage: a(0) 1 sage: a(1) 1 sage: a(2) 3 sage: a(3) 7 sage: a(11) 8119 sage: a.list(12) [1, 1, 3, 7, 17, 41, 99, 239, 577, 1393, 3363, 8119] - >>> from sage.all import * >>> a = sloane.A001333;a Numerators of continued fraction convergents to sqrt(2). >>> a(Integer(0)) 1 >>> a(Integer(1)) 1 >>> a(Integer(2)) 3 >>> a(Integer(3)) 7 >>> a(Integer(11)) 8119 >>> a.list(Integer(12)) [1, 1, 3, 7, 17, 41, 99, 239, 577, 1393, 3363, 8119] - AUTHORS: - Jaap Spies (2007-02-01) 
 
- class sage.combinat.sloane_functions.A001358[source]¶
- Bases: - SloaneSequence- Products of two primes. - These numbers have been called semiprimes (or semi-primes), biprimes or 2-almost primes. - INPUT: - n– nonnegative integer
 - OUTPUT: integer; function value - EXAMPLES: - sage: a = sloane.A001358;a Products of two primes. sage: a(0) Traceback (most recent call last): ... ValueError: input n (=0) must be a positive integer sage: a(2) 6 sage: a(8) 22 sage: a(200) 669 sage: a.list(9) [4, 6, 9, 10, 14, 15, 21, 22, 25] - >>> from sage.all import * >>> a = sloane.A001358;a Products of two primes. >>> a(Integer(0)) Traceback (most recent call last): ... ValueError: input n (=0) must be a positive integer >>> a(Integer(2)) 6 >>> a(Integer(8)) 22 >>> a(Integer(200)) 669 >>> a.list(Integer(9)) [4, 6, 9, 10, 14, 15, 21, 22, 25] - AUTHORS: - Jaap Spies (2007-01-25) 
 
- class sage.combinat.sloane_functions.A001405[source]¶
- Bases: - SloaneSequence- Central binomial coefficients: \(\binom{n}{\lfloor \frac {n}{ 2} \rfloor}\). - INPUT: - n– nonnegative integer
 - OUTPUT: integer; function value - EXAMPLES: - sage: a = sloane.A001405;a Central binomial coefficients: C(n,floor(n/2)). sage: a(0) 1 sage: a(2) 2 sage: a(12) 924 sage: a.list(12) [1, 1, 2, 3, 6, 10, 20, 35, 70, 126, 252, 462] - >>> from sage.all import * >>> a = sloane.A001405;a Central binomial coefficients: C(n,floor(n/2)). >>> a(Integer(0)) 1 >>> a(Integer(2)) 2 >>> a(Integer(12)) 924 >>> a.list(Integer(12)) [1, 1, 2, 3, 6, 10, 20, 35, 70, 126, 252, 462] - AUTHORS: - Jaap Spies (2007-01-26) 
 
- class sage.combinat.sloane_functions.A001477[source]¶
- Bases: - SloaneSequence- The nonnegative integers. - INPUT: - n– nonnegative integer
 - OUTPUT: integer; function value - EXAMPLES: - sage: a = sloane.A001477;a The nonnegative integers. sage: a(-1) Traceback (most recent call last): ... ValueError: input n (=-1) must be an integer >= 0 sage: a(0) 0 sage: a(3382789) 3382789 sage: a(11) 11 sage: a.list(12) [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11] - >>> from sage.all import * >>> a = sloane.A001477;a The nonnegative integers. >>> a(-Integer(1)) Traceback (most recent call last): ... ValueError: input n (=-1) must be an integer >= 0 >>> a(Integer(0)) 0 >>> a(Integer(3382789)) 3382789 >>> a(Integer(11)) 11 >>> a.list(Integer(12)) [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11] - AUTHORS: - Jaap Spies (2007-01-26) 
 
- class sage.combinat.sloane_functions.A001694[source]¶
- Bases: - SloaneSequence- This function returns the \(n\)-th Powerful Number: - A positive integer \(n\) is powerful if for every prime \(p\) dividing \(n\), \(p^2\) also divides \(n\). - INPUT: - n– nonnegative integer
 - OUTPUT: integer; function value - EXAMPLES: - sage: a = sloane.A001694; a Powerful Numbers (also called squarefull, square-full or 2-full numbers). sage: a.offset 1 sage: a(1) 1 sage: a(4) 9 sage: a(100) 3136 sage: a(156) 7225 sage: a.list(19) [1, 4, 8, 9, 16, 25, 27, 32, 36, 49, 64, 72, 81, 100, 108, 121, 125, 128, 144] sage: a(-1) Traceback (most recent call last): ... ValueError: input n (=-1) must be a positive integer - >>> from sage.all import * >>> a = sloane.A001694; a Powerful Numbers (also called squarefull, square-full or 2-full numbers). >>> a.offset 1 >>> a(Integer(1)) 1 >>> a(Integer(4)) 9 >>> a(Integer(100)) 3136 >>> a(Integer(156)) 7225 >>> a.list(Integer(19)) [1, 4, 8, 9, 16, 25, 27, 32, 36, 49, 64, 72, 81, 100, 108, 121, 125, 128, 144] >>> a(-Integer(1)) Traceback (most recent call last): ... ValueError: input n (=-1) must be a positive integer - AUTHORS: - Jaap Spies (2007-01-14) 
 - is_powerful(n)[source]¶
- Return - Trueif and only if \(n\) is a powerful number.- A positive integer \(n\) is powerful if for every prime \(p\) dividing \(n\), \(p^2\) also divides \(n\). - INPUT: - n– integer
 - OUTPUT: - Trueif \(n\) is a powerful number, else- False- EXAMPLES: - sage: a = sloane.A001694 sage: a.is_powerful(2500) True sage: a.is_powerful(20) False - >>> from sage.all import * >>> a = sloane.A001694 >>> a.is_powerful(Integer(2500)) True >>> a.is_powerful(Integer(20)) False - AUTHORS: - Jaap Spies (2006-12-07) 
 
 
- class sage.combinat.sloane_functions.A001836[source]¶
- Bases: - SloaneSequence- Numbers \(n\) such that \(\phi(2n-1) < \phi(2n)\), where \(\phi\) is Euler’s totient function. - Euler’s totient function is also known as - euler_phi;- euler_phiis a standard Sage function.- INPUT: - n– nonnegative integer
 - OUTPUT: integer; function value - EXAMPLES: - sage: a = sloane.A001836; a Numbers n such that phi(2n-1) < phi(2n), where phi is Euler's totient function A000010. sage: a.offset 1 sage: a(1) 53 sage: a(8) 683 sage: a(300) 17798 sage: a.list(12) [53, 83, 158, 263, 293, 368, 578, 683, 743, 788, 878, 893] sage: a(0) Traceback (most recent call last): ... ValueError: input n (=0) must be a positive integer - >>> from sage.all import * >>> a = sloane.A001836; a Numbers n such that phi(2n-1) < phi(2n), where phi is Euler's totient function A000010. >>> a.offset 1 >>> a(Integer(1)) 53 >>> a(Integer(8)) 683 >>> a(Integer(300)) 17798 >>> a.list(Integer(12)) [53, 83, 158, 263, 293, 368, 578, 683, 743, 788, 878, 893] >>> a(Integer(0)) Traceback (most recent call last): ... ValueError: input n (=0) must be a positive integer - Compare: Searching Sloane’s online database… Numbers n such that phi(2n-1) phi(2n), where phi is Euler’s totient function A000010. [53, 83, 158, 263, 293, 368, 578, 683, 743, 788, 878, 893] - AUTHORS: - Jaap Spies (2007-01-17) 
 
- class sage.combinat.sloane_functions.A001906[source]¶
- Bases: - RecurrenceSequence2- \(F(2n) =\) bisection of Fibonacci sequence: \(a(n)=3a(n-1)-a(n-2)\). - INPUT: - n– nonnegative integer
 - OUTPUT: integer; function value - EXAMPLES: - sage: a = sloane.A001906; a F(2n) = bisection of Fibonacci sequence: a(n)=3a(n-1)-a(n-2). sage: a(0) 0 sage: a(1) 1 sage: a(8) 987 sage: a(22) 701408733 sage: a.list(12) [0, 1, 3, 8, 21, 55, 144, 377, 987, 2584, 6765, 17711] - >>> from sage.all import * >>> a = sloane.A001906; a F(2n) = bisection of Fibonacci sequence: a(n)=3a(n-1)-a(n-2). >>> a(Integer(0)) 0 >>> a(Integer(1)) 1 >>> a(Integer(8)) 987 >>> a(Integer(22)) 701408733 >>> a.list(Integer(12)) [0, 1, 3, 8, 21, 55, 144, 377, 987, 2584, 6765, 17711] - AUTHORS: - Jaap Spies (2007-01-19) 
 
- class sage.combinat.sloane_functions.A001909[source]¶
- Bases: - ExtremesOfPermanentsSequence- \(a(n) = n*a(n-1) + (n-4)*a(n-2)\), with \(a(2) = 0\), \(a(3) = 1\). - With offset 1, permanent of (0,1)-matrix of size \(n \times (n+d)\) with \(d=4\) and \(n\) zeros not on a line. This is a special case of Theorem 2.3 of Seok-Zun Song et al. Extremes of permanents of (0,1)-matrices, p. 201-202. - Seok-Zun Song et al., Extremes of permanents of (0,1)-matrices, Lin. Algebra and its Applic. 373 (2003), p. 197-210. - INPUT: - n– positive integer >= 2
 - OUTPUT: integer; function value - EXAMPLES: - sage: a = sloane.A001909;a a(n) = n*a(n-1) + (n-4)*a(n-2), a(2) = 0, a(3) = 1. sage: a(1) Traceback (most recent call last): ... ValueError: input n (=1) must be an integer >= 2 sage: a.offset 2 sage: a(2) 0 sage: a(8) 8544 sage: a(22) 470033715095287415734 sage: a.list(9) [0, 1, 4, 21, 134, 1001, 8544, 81901, 870274] - >>> from sage.all import * >>> a = sloane.A001909;a a(n) = n*a(n-1) + (n-4)*a(n-2), a(2) = 0, a(3) = 1. >>> a(Integer(1)) Traceback (most recent call last): ... ValueError: input n (=1) must be an integer >= 2 >>> a.offset 2 >>> a(Integer(2)) 0 >>> a(Integer(8)) 8544 >>> a(Integer(22)) 470033715095287415734 >>> a.list(Integer(9)) [0, 1, 4, 21, 134, 1001, 8544, 81901, 870274] - AUTHORS: - Jaap Spies (2007-01-13) 
 
- class sage.combinat.sloane_functions.A001910[source]¶
- Bases: - ExtremesOfPermanentsSequence- \(a(n) = n*a(n-1) + (n-5)*a(n-2)\), with \(a(3) = 0\), \(a(4) = 1\). - With offset 1, permanent of (0,1)-matrix of size \(n \times (n+d)\) with \(d=5\) and \(n\) zeros not on a line. This is a special case of Theorem 2.3 of Seok-Zun Song et al. Extremes of permanents of (0,1)-matrices, p. 201-202. - Seok-Zun Song et al., Extremes of permanents of (0,1)-matrices, Lin. Algebra and its Applic. 373 (2003), p. 197-210. - INPUT: - n– positive integer \(\geq 3\)
 - OUTPUT: integer; function value - EXAMPLES: - sage: a = sloane.A001910;a a(n) = n*a(n-1) + (n-5)*a(n-2), a(3) = 0, a(4) = 1. sage: a(0) Traceback (most recent call last): ... ValueError: input n (=0) must be an integer >= 3 sage: a(3) 0 sage: a.offset 3 sage: a(8) 1909 sage: a(22) 98125321641110663023 sage: a.list(9) [0, 1, 5, 31, 227, 1909, 18089, 190435, 2203319] - >>> from sage.all import * >>> a = sloane.A001910;a a(n) = n*a(n-1) + (n-5)*a(n-2), a(3) = 0, a(4) = 1. >>> a(Integer(0)) Traceback (most recent call last): ... ValueError: input n (=0) must be an integer >= 3 >>> a(Integer(3)) 0 >>> a.offset 3 >>> a(Integer(8)) 1909 >>> a(Integer(22)) 98125321641110663023 >>> a.list(Integer(9)) [0, 1, 5, 31, 227, 1909, 18089, 190435, 2203319] - AUTHORS: - Jaap Spies (2007-01-13) 
 
- class sage.combinat.sloane_functions.A001969[source]¶
- Bases: - SloaneSequence- Evil numbers: even number of 1s in binary expansion. - INPUT: - n– nonnegative integer
 - OUTPUT: integer; function value - EXAMPLES: - sage: a = sloane.A001969;a Evil numbers: even number of 1s in binary expansion. sage: a(0) 0 sage: a(1) 3 sage: a(2) 5 sage: a(12) 24 sage: a.list(13) [0, 3, 5, 6, 9, 10, 12, 15, 17, 18, 20, 23, 24] - >>> from sage.all import * >>> a = sloane.A001969;a Evil numbers: even number of 1s in binary expansion. >>> a(Integer(0)) 0 >>> a(Integer(1)) 3 >>> a(Integer(2)) 5 >>> a(Integer(12)) 24 >>> a.list(Integer(13)) [0, 3, 5, 6, 9, 10, 12, 15, 17, 18, 20, 23, 24] - AUTHORS: - Jaap Spies (2007-02-02) 
 
- class sage.combinat.sloane_functions.A002110[source]¶
- Bases: - SloaneSequence- Primorial numbers (first definition): product of first \(n\) primes. Sometimes written \(p\#\). - INPUT: - n– nonnegative integer
 - OUTPUT: integer; function value - EXAMPLES: - sage: a = sloane.A002110;a Primorial numbers (first definition): product of first n primes. Sometimes written p#. sage: a(0) 1 sage: a(2) 6 sage: a(8) 9699690 sage: a(17) 1922760350154212639070 sage: a.list(9) [1, 2, 6, 30, 210, 2310, 30030, 510510, 9699690] - >>> from sage.all import * >>> a = sloane.A002110;a Primorial numbers (first definition): product of first n primes. Sometimes written p#. >>> a(Integer(0)) 1 >>> a(Integer(2)) 6 >>> a(Integer(8)) 9699690 >>> a(Integer(17)) 1922760350154212639070 >>> a.list(Integer(9)) [1, 2, 6, 30, 210, 2310, 30030, 510510, 9699690] - AUTHORS: - Jaap Spies (2007-01-25) 
 
- class sage.combinat.sloane_functions.A002113[source]¶
- Bases: - SloaneSequence- Palindromes in base 10. - INPUT: - n– nonnegative integer
 - OUTPUT: integer; function value - EXAMPLES: - sage: a = sloane.A002113;a Palindromes in base 10. sage: a(0) 0 sage: a(1) 1 sage: a(2) 2 sage: a(12) 33 sage: a.list(13) [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 22, 33] - >>> from sage.all import * >>> a = sloane.A002113;a Palindromes in base 10. >>> a(Integer(0)) 0 >>> a(Integer(1)) 1 >>> a(Integer(2)) 2 >>> a(Integer(12)) 33 >>> a.list(Integer(13)) [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 22, 33] - AUTHORS: - Jaap Spies (2007-02-02) 
 
- class sage.combinat.sloane_functions.A002275[source]¶
- Bases: - SloaneSequence- Repunits: \(\frac {(10^n - 1)}{9}\). Often denoted by \(R_n\). - INPUT: - n– nonnegative integer
 - OUTPUT: integer; function value - EXAMPLES: - sage: a = sloane.A002275;a Repunits: (10^n - 1)/9. Often denoted by R_n. sage: a(0) 0 sage: a(2) 11 sage: a(8) 11111111 sage: a(20) 11111111111111111111 sage: a.list(9) [0, 1, 11, 111, 1111, 11111, 111111, 1111111, 11111111] - >>> from sage.all import * >>> a = sloane.A002275;a Repunits: (10^n - 1)/9. Often denoted by R_n. >>> a(Integer(0)) 0 >>> a(Integer(2)) 11 >>> a(Integer(8)) 11111111 >>> a(Integer(20)) 11111111111111111111 >>> a.list(Integer(9)) [0, 1, 11, 111, 1111, 11111, 111111, 1111111, 11111111] - AUTHORS: - Jaap Spies (2007-01-25) 
 
- class sage.combinat.sloane_functions.A002378[source]¶
- Bases: - SloaneSequence- Oblong (or pronic, or heteromecic) numbers: \(n(n+1)\). - INPUT: - n– nonnegative integer
 - OUTPUT: integer; function value - EXAMPLES: - sage: a = sloane.A002378;a Oblong (or pronic, or heteromecic) numbers: n(n+1). sage: a(-1) Traceback (most recent call last): ... ValueError: input n (=-1) must be an integer >= 0 sage: a(0) 0 sage: a(1) 2 sage: a(11) 132 sage: a.list(12) [0, 2, 6, 12, 20, 30, 42, 56, 72, 90, 110, 132] - >>> from sage.all import * >>> a = sloane.A002378;a Oblong (or pronic, or heteromecic) numbers: n(n+1). >>> a(-Integer(1)) Traceback (most recent call last): ... ValueError: input n (=-1) must be an integer >= 0 >>> a(Integer(0)) 0 >>> a(Integer(1)) 2 >>> a(Integer(11)) 132 >>> a.list(Integer(12)) [0, 2, 6, 12, 20, 30, 42, 56, 72, 90, 110, 132] - AUTHORS: - Jaap Spies (2007-01-26) 
 
- class sage.combinat.sloane_functions.A002620[source]¶
- Bases: - SloaneSequence- Quarter-squares: floor(n/2)*ceiling(n/2). Equivalently, \(\lfloor n^2/4 \rfloor\). - INPUT: - n– nonnegative integer
 - OUTPUT: integer; function value - EXAMPLES: - sage: a = sloane.A002620;a Quarter-squares: floor(n/2)*ceiling(n/2). Equivalently, floor(n^2/4). sage: a(0) 0 sage: a(1) 0 sage: a(2) 1 sage: a(10) 25 sage: a.list(12) [0, 0, 1, 2, 4, 6, 9, 12, 16, 20, 25, 30] - >>> from sage.all import * >>> a = sloane.A002620;a Quarter-squares: floor(n/2)*ceiling(n/2). Equivalently, floor(n^2/4). >>> a(Integer(0)) 0 >>> a(Integer(1)) 0 >>> a(Integer(2)) 1 >>> a(Integer(10)) 25 >>> a.list(Integer(12)) [0, 0, 1, 2, 4, 6, 9, 12, 16, 20, 25, 30] - AUTHORS: - Jaap Spies (2007-01-26) 
 
- class sage.combinat.sloane_functions.A002808[source]¶
- Bases: - SloaneSequence- The composite numbers: numbers \(n\) of the form \(xy\) for \(x > 1\) and \(y > 1\). - INPUT: - n– nonnegative integer
 - OUTPUT: integer; function value - EXAMPLES: - sage: a = sloane.A002808;a The composite numbers: numbers n of the form x*y for x > 1 and y > 1. sage: a(0) Traceback (most recent call last): ... ValueError: input n (=0) must be a positive integer sage: a(2) 6 sage: a(11) 20 sage: a.list(12) [4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21] - >>> from sage.all import * >>> a = sloane.A002808;a The composite numbers: numbers n of the form x*y for x > 1 and y > 1. >>> a(Integer(0)) Traceback (most recent call last): ... ValueError: input n (=0) must be a positive integer >>> a(Integer(2)) 6 >>> a(Integer(11)) 20 >>> a.list(Integer(12)) [4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21] - AUTHORS: - Jaap Spies (2007-01-26) 
 
- class sage.combinat.sloane_functions.A003418[source]¶
- Bases: - SloaneSequence- Least common multiple (or lcm) of \(\{1, 2, \ldots, n\}\). - INPUT: - n– nonnegative integer
 - OUTPUT: integer; function value - EXAMPLES: - sage: a = sloane.A003418;a Least common multiple (or lcm) of {1, 2, ..., n}. sage: a(0) 1 sage: a(1) 1 sage: a(13) 360360 sage: a.list(14) [1, 1, 2, 6, 12, 60, 60, 420, 840, 2520, 2520, 27720, 27720, 360360] sage: a(20.0) Traceback (most recent call last): ... TypeError: input must be an int or Integer - >>> from sage.all import * >>> a = sloane.A003418;a Least common multiple (or lcm) of {1, 2, ..., n}. >>> a(Integer(0)) 1 >>> a(Integer(1)) 1 >>> a(Integer(13)) 360360 >>> a.list(Integer(14)) [1, 1, 2, 6, 12, 60, 60, 420, 840, 2520, 2520, 27720, 27720, 360360] >>> a(RealNumber('20.0')) Traceback (most recent call last): ... TypeError: input must be an int or Integer - AUTHOR: - Jaap Spies (2007-01-31) 
 
- class sage.combinat.sloane_functions.A004086[source]¶
- Bases: - SloaneSequence- Read n backwards (referred to as \(R(n)\) in many sequences). - INPUT: - n– nonnegative integer
 - OUTPUT: integer; function value - EXAMPLES: - sage: a = sloane.A004086;a Read n backwards (referred to as R(n) in many sequences). sage: a(0) 0 sage: a(1) 1 sage: a(2) 2 sage: a(3333) 3333 sage: a(12345) 54321 sage: a.list(13) [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 11, 21] - >>> from sage.all import * >>> a = sloane.A004086;a Read n backwards (referred to as R(n) in many sequences). >>> a(Integer(0)) 0 >>> a(Integer(1)) 1 >>> a(Integer(2)) 2 >>> a(Integer(3333)) 3333 >>> a(Integer(12345)) 54321 >>> a.list(Integer(13)) [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 11, 21] - AUTHORS: - Jaap Spies (2007-02-02) 
 
- class sage.combinat.sloane_functions.A004526[source]¶
- Bases: - SloaneSequence- The nonnegative integers repeated. - INPUT: - n– nonnegative integer
 - OUTPUT: integer; function value - EXAMPLES: - sage: a = sloane.A004526;a The nonnegative integers repeated. sage: a(0) 0 sage: a(1) 0 sage: a(2) 1 sage: a(10) 5 sage: a.list(12) [0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5] - >>> from sage.all import * >>> a = sloane.A004526;a The nonnegative integers repeated. >>> a(Integer(0)) 0 >>> a(Integer(1)) 0 >>> a(Integer(2)) 1 >>> a(Integer(10)) 5 >>> a.list(Integer(12)) [0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5] - AUTHORS: - Jaap Spies (2007-01-26) 
 
- class sage.combinat.sloane_functions.A005100[source]¶
- Bases: - SloaneSequence- Deficient numbers: \(\sigma(n) < 2n\). - INPUT: - n– nonnegative integer
 - OUTPUT: integer; function value - EXAMPLES: - sage: a = sloane.A005100;a Deficient numbers: sigma(n) < 2n sage: a(0) Traceback (most recent call last): ... ValueError: input n (=0) must be a positive integer sage: a(1) 1 sage: a(2) 2 sage: a(12) 14 sage: a.list(12) [1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 13, 14] - >>> from sage.all import * >>> a = sloane.A005100;a Deficient numbers: sigma(n) < 2n >>> a(Integer(0)) Traceback (most recent call last): ... ValueError: input n (=0) must be a positive integer >>> a(Integer(1)) 1 >>> a(Integer(2)) 2 >>> a(Integer(12)) 14 >>> a.list(Integer(12)) [1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 13, 14] - AUTHORS: - Jaap Spies (2007-01-26) 
 
- class sage.combinat.sloane_functions.A005101[source]¶
- Bases: - SloaneSequence- Abundant numbers (sum of divisors of \(n\) exceeds \(2n\)). - INPUT: - n– nonnegative integer
 - OUTPUT: integer; function value - EXAMPLES: - sage: a = sloane.A005101;a Abundant numbers (sum of divisors of n exceeds 2n). sage: a(0) Traceback (most recent call last): ... ValueError: input n (=0) must be a positive integer sage: a(1) 12 sage: a(2) 18 sage: a(12) 60 sage: a.list(12) [12, 18, 20, 24, 30, 36, 40, 42, 48, 54, 56, 60] - >>> from sage.all import * >>> a = sloane.A005101;a Abundant numbers (sum of divisors of n exceeds 2n). >>> a(Integer(0)) Traceback (most recent call last): ... ValueError: input n (=0) must be a positive integer >>> a(Integer(1)) 12 >>> a(Integer(2)) 18 >>> a(Integer(12)) 60 >>> a.list(Integer(12)) [12, 18, 20, 24, 30, 36, 40, 42, 48, 54, 56, 60] - AUTHORS: - Jaap Spies (2007-01-26) 
 
- class sage.combinat.sloane_functions.A005117[source]¶
- Bases: - SloaneSequence- Square-free numbers. - INPUT: - n– nonnegative integer
 - OUTPUT: integer; function value - EXAMPLES: - sage: a = sloane.A005117;a Square-free numbers. sage: a(0) Traceback (most recent call last): ... ValueError: input n (=0) must be a positive integer sage: a(2) 2 sage: a(12) 17 sage: a.list(12) [1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 17] - >>> from sage.all import * >>> a = sloane.A005117;a Square-free numbers. >>> a(Integer(0)) Traceback (most recent call last): ... ValueError: input n (=0) must be a positive integer >>> a(Integer(2)) 2 >>> a(Integer(12)) 17 >>> a.list(Integer(12)) [1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 17] - AUTHORS: - Jaap Spies (2007-01-25) 
 
- class sage.combinat.sloane_functions.A005408[source]¶
- Bases: - SloaneSequence- The odd numbers a(n) = 2n + 1. - INPUT: - n– nonnegative integer
 - OUTPUT: integer; function value - EXAMPLES: - sage: a = sloane.A005408;a The odd numbers a(n) = 2n + 1. sage: a(-1) Traceback (most recent call last): ... ValueError: input n (=-1) must be an integer >= 0 sage: a(0) 1 sage: a(4) 9 sage: a(11) 23 sage: a.list(12) [1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23] - >>> from sage.all import * >>> a = sloane.A005408;a The odd numbers a(n) = 2n + 1. >>> a(-Integer(1)) Traceback (most recent call last): ... ValueError: input n (=-1) must be an integer >= 0 >>> a(Integer(0)) 1 >>> a(Integer(4)) 9 >>> a(Integer(11)) 23 >>> a.list(Integer(12)) [1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23] - AUTHORS: - Jaap Spies (2007-01-26) 
 
- class sage.combinat.sloane_functions.A005843[source]¶
- Bases: - SloaneSequence- The even numbers: \(a(n) = 2n\). - INPUT: - n– nonnegative integer
 - OUTPUT: integer; function value - EXAMPLES: - sage: a = sloane.A005843;a The even numbers: a(n) = 2n. sage: a(0.0) Traceback (most recent call last): ... TypeError: input must be an int or Integer sage: a(1) 2 sage: a(2) 4 sage: a(9) 18 sage: a.list(10) [0, 2, 4, 6, 8, 10, 12, 14, 16, 18] - >>> from sage.all import * >>> a = sloane.A005843;a The even numbers: a(n) = 2n. >>> a(RealNumber('0.0')) Traceback (most recent call last): ... TypeError: input must be an int or Integer >>> a(Integer(1)) 2 >>> a(Integer(2)) 4 >>> a(Integer(9)) 18 >>> a.list(Integer(10)) [0, 2, 4, 6, 8, 10, 12, 14, 16, 18] - AUTHORS: - Jaap Spies (2007-02-03) 
 
- class sage.combinat.sloane_functions.A006318[source]¶
- Bases: - SloaneSequence- Large Schroeder numbers. - INPUT: - n– nonnegative integer
 - OUTPUT: integer; function value - EXAMPLES: - sage: a = sloane.A006318;a Large Schroeder numbers. sage: a(0) 1 sage: a(1) 2 sage: a(2) 6 sage: a(9) 206098 sage: a.list(10) [1, 2, 6, 22, 90, 394, 1806, 8558, 41586, 206098] - >>> from sage.all import * >>> a = sloane.A006318;a Large Schroeder numbers. >>> a(Integer(0)) 1 >>> a(Integer(1)) 2 >>> a(Integer(2)) 6 >>> a(Integer(9)) 206098 >>> a.list(Integer(10)) [1, 2, 6, 22, 90, 394, 1806, 8558, 41586, 206098] - AUTHORS: - Jaap Spies (2007-02-03) 
 
- class sage.combinat.sloane_functions.A006530[source]¶
- Bases: - SloaneSequence- Largest prime dividing \(n\) (with \(a(1)=1\)). - INPUT: - n– nonnegative integer
 - OUTPUT: integer; function value - EXAMPLES: - sage: a = sloane.A006530;a Largest prime dividing n (with a(1)=1). sage: a(0) Traceback (most recent call last): ... ValueError: input n (=0) must be a positive integer sage: a(1) 1 sage: a(2) 2 sage: a(8) 2 sage: a(11) 11 sage: a.list(15) [1, 2, 3, 2, 5, 3, 7, 2, 3, 5, 11, 3, 13, 7, 5] - >>> from sage.all import * >>> a = sloane.A006530;a Largest prime dividing n (with a(1)=1). >>> a(Integer(0)) Traceback (most recent call last): ... ValueError: input n (=0) must be a positive integer >>> a(Integer(1)) 1 >>> a(Integer(2)) 2 >>> a(Integer(8)) 2 >>> a(Integer(11)) 11 >>> a.list(Integer(15)) [1, 2, 3, 2, 5, 3, 7, 2, 3, 5, 11, 3, 13, 7, 5] - AUTHORS: - Jaap Spies (2007-01-25) 
 
- class sage.combinat.sloane_functions.A006882[source]¶
- Bases: - SloaneSequence- Double factorials \(n!!\): \(a(n)=n \cdot a(n-2)\). - INPUT: - n– nonnegative integer
 - OUTPUT: integer; function value - EXAMPLES: - sage: a = sloane.A006882;a Double factorials n!!: a(n)=n*a(n-2). sage: a(0) 1 sage: a(2) 2 sage: a(8) 384 sage: a(20) 3715891200 sage: a.list(9) [1, 1, 2, 3, 8, 15, 48, 105, 384] - >>> from sage.all import * >>> a = sloane.A006882;a Double factorials n!!: a(n)=n*a(n-2). >>> a(Integer(0)) 1 >>> a(Integer(2)) 2 >>> a(Integer(8)) 384 >>> a(Integer(20)) 3715891200 >>> a.list(Integer(9)) [1, 1, 2, 3, 8, 15, 48, 105, 384] - AUTHORS: - Jaap Spies (2007-01-24) 
 - df()[source]¶
- Double factorials n!!: a(n)=n*a(n-2). - EXAMPLES: - sage: it = sloane.A006882.df() sage: [next(it) for i in range(10)] [1, 1, 2, 3, 8, 15, 48, 105, 384, 945] - >>> from sage.all import * >>> it = sloane.A006882.df() >>> [next(it) for i in range(Integer(10))] [1, 1, 2, 3, 8, 15, 48, 105, 384, 945] 
 
- class sage.combinat.sloane_functions.A007318[source]¶
- Bases: - SloaneSequence- Pascal’s triangle read by rows: \(C(n,k) = \binom{n}{k} = \frac {n!} {(k!(n-k)!)}\), \(0 \le k \le n\). - INPUT: - n– nonnegative integer
 - OUTPUT: integer; function value - EXAMPLES: - sage: a = sloane.A007318 sage: a(0) 1 sage: a(1) 1 sage: a(13) 4 sage: a.list(15) [1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 6, 4, 1] sage: a(100) 715 - >>> from sage.all import * >>> a = sloane.A007318 >>> a(Integer(0)) 1 >>> a(Integer(1)) 1 >>> a(Integer(13)) 4 >>> a.list(Integer(15)) [1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 6, 4, 1] >>> a(Integer(100)) 715 - AUTHORS: - Jaap Spies (2007-01-31) 
 - keyword = ['nonn', 'tabl', 'nice', 'easy', 'core', 'triangle']¶
 
- class sage.combinat.sloane_functions.A008275[source]¶
- Bases: - SloaneSequence- Triangle of Stirling numbers of first kind, \(s(n,k)\), \(n \ge 1\), \(1 \le k \le n\). - The unsigned numbers are also called Stirling cycle numbers: - \(|s(n,k)|\) = number of permutations of \(n\) objects with exactly \(k\) cycles. - INPUT: - n– nonnegative integer
 - OUTPUT: integer; function value - EXAMPLES: - sage: a = sloane.A008275;a Triangle of Stirling numbers of first kind, s(n,k), n >= 1, 1<=k<=n. sage: a(0) Traceback (most recent call last): ... ValueError: input n (=0) must be a positive integer sage: a(1) 1 sage: a(2) -1 sage: a(3) 1 sage: a(11) 24 sage: a.list(12) [1, -1, 1, 2, -3, 1, -6, 11, -6, 1, 24, -50] - >>> from sage.all import * >>> a = sloane.A008275;a Triangle of Stirling numbers of first kind, s(n,k), n >= 1, 1<=k<=n. >>> a(Integer(0)) Traceback (most recent call last): ... ValueError: input n (=0) must be a positive integer >>> a(Integer(1)) 1 >>> a(Integer(2)) -1 >>> a(Integer(3)) 1 >>> a(Integer(11)) 24 >>> a.list(Integer(12)) [1, -1, 1, 2, -3, 1, -6, 11, -6, 1, 24, -50] - AUTHORS: - Jaap Spies (2007-02-02) 
 - keyword = ['sign', 'tabl', 'nice', 'core', 'triangle']¶
 
- class sage.combinat.sloane_functions.A008277[source]¶
- Bases: - SloaneSequence- Triangle of Stirling numbers of 2nd kind, \(S2(n,k)\), \(n \ge 1\), \(1 \le k \le n\). - INPUT: - n– nonnegative integer
 - OUTPUT: integer; function value - EXAMPLES: - sage: a = sloane.A008277;a Triangle of Stirling numbers of 2nd kind, S2(n,k), n >= 1, 1<=k<=n. sage: a(0) Traceback (most recent call last): ... ValueError: input n (=0) must be a positive integer sage: a(1) 1 sage: a(2) 1 sage: a(3) 1 sage: a(4.0) Traceback (most recent call last): ... TypeError: input must be an int or Integer sage: a.list(15) [1, 1, 1, 1, 3, 1, 1, 7, 6, 1, 1, 15, 25, 10, 1] - >>> from sage.all import * >>> a = sloane.A008277;a Triangle of Stirling numbers of 2nd kind, S2(n,k), n >= 1, 1<=k<=n. >>> a(Integer(0)) Traceback (most recent call last): ... ValueError: input n (=0) must be a positive integer >>> a(Integer(1)) 1 >>> a(Integer(2)) 1 >>> a(Integer(3)) 1 >>> a(RealNumber('4.0')) Traceback (most recent call last): ... TypeError: input must be an int or Integer >>> a.list(Integer(15)) [1, 1, 1, 1, 3, 1, 1, 7, 6, 1, 1, 15, 25, 10, 1] - AUTHORS: - Jaap Spies (2007-01-31) 
 - keyword = ['nonn', 'tabl', 'nice', 'core', 'triangle']¶
 
- class sage.combinat.sloane_functions.A008683[source]¶
- Bases: - SloaneSequence- Möbius function \(\mu(n)\). - INPUT: - n– nonnegative integer
 - OUTPUT: integer; function value - EXAMPLES: - sage: a = sloane.A008683;a Moebius function mu(n). sage: a(0) Traceback (most recent call last): ... ValueError: input n (=0) must be a positive integer sage: a(2) -1 sage: a(12) 0 sage: a.list(12) [1, -1, -1, 0, -1, 1, -1, 0, 0, 1, -1, 0] - >>> from sage.all import * >>> a = sloane.A008683;a Moebius function mu(n). >>> a(Integer(0)) Traceback (most recent call last): ... ValueError: input n (=0) must be a positive integer >>> a(Integer(2)) -1 >>> a(Integer(12)) 0 >>> a.list(Integer(12)) [1, -1, -1, 0, -1, 1, -1, 0, 0, 1, -1, 0] - AUTHORS: - Jaap Spies (2007-01-13) 
 
- class sage.combinat.sloane_functions.A010060[source]¶
- Bases: - SloaneSequence- Thue-Morse sequence. - Let \(A_k\) denote the first \(2^k\) terms; then \(A_0 = 0\), and for \(k \ge 0\), \(A_{k+1} = A_k B_k\), where \(B_k\) is obtained from \(A_k\) by interchanging 0s and 1s. - INPUT: - n– nonnegative integer
 - OUTPUT: integer; function value - EXAMPLES: - sage: a = sloane.A010060;a Thue-Morse sequence. sage: a(0) 0 sage: a(1) 1 sage: a(2) 1 sage: a(12) 0 sage: a.list(13) [0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0] - >>> from sage.all import * >>> a = sloane.A010060;a Thue-Morse sequence. >>> a(Integer(0)) 0 >>> a(Integer(1)) 1 >>> a(Integer(2)) 1 >>> a(Integer(12)) 0 >>> a.list(Integer(13)) [0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0] - AUTHORS: - Jaap Spies (2007-02-02) 
 
- class sage.combinat.sloane_functions.A015521[source]¶
- Bases: - RecurrenceSequence2- Linear 2nd order recurrence, \(a(0)=0\), \(a(1)=1\) and \(a(n) = 3 a(n-1) + 4 a(n-2)\). - INPUT: - n– nonnegative integer
 - OUTPUT: integer; function value - EXAMPLES: - sage: a = sloane.A015521; a Linear 2nd order recurrence, a(n) = 3 a(n-1) + 4 a(n-2). sage: a(0) 0 sage: a(1) 1 sage: a(8) 13107 sage: a(41) 967140655691703339764941 sage: a.list(12) [0, 1, 3, 13, 51, 205, 819, 3277, 13107, 52429, 209715, 838861] - >>> from sage.all import * >>> a = sloane.A015521; a Linear 2nd order recurrence, a(n) = 3 a(n-1) + 4 a(n-2). >>> a(Integer(0)) 0 >>> a(Integer(1)) 1 >>> a(Integer(8)) 13107 >>> a(Integer(41)) 967140655691703339764941 >>> a.list(Integer(12)) [0, 1, 3, 13, 51, 205, 819, 3277, 13107, 52429, 209715, 838861] - AUTHORS: - Jaap Spies (2007-01-19) 
 
- class sage.combinat.sloane_functions.A015523[source]¶
- Bases: - RecurrenceSequence2- Linear 2nd order recurrence, \(a(0)=0\), \(a(1)=1\) and \(a(n) = 3 a(n-1) + 5 a(n-2)\). - INPUT: - n– nonnegative integer
 - OUTPUT: integer; function value - EXAMPLES: - sage: a = sloane.A015523; a Linear 2nd order recurrence, a(n) = 3 a(n-1) + 5 a(n-2). sage: a(0) 0 sage: a(1) 1 sage: a(8) 17727 sage: a(41) 6173719566474529739091481 sage: a.list(12) [0, 1, 3, 14, 57, 241, 1008, 4229, 17727, 74326, 311613, 1306469] - >>> from sage.all import * >>> a = sloane.A015523; a Linear 2nd order recurrence, a(n) = 3 a(n-1) + 5 a(n-2). >>> a(Integer(0)) 0 >>> a(Integer(1)) 1 >>> a(Integer(8)) 17727 >>> a(Integer(41)) 6173719566474529739091481 >>> a.list(Integer(12)) [0, 1, 3, 14, 57, 241, 1008, 4229, 17727, 74326, 311613, 1306469] - AUTHORS: - Jaap Spies (2007-01-19) 
 
- class sage.combinat.sloane_functions.A015530[source]¶
- Bases: - RecurrenceSequence2- Linear 2nd order recurrence, \(a(0)=0\), \(a(1)=1\) and \(a(n) = 4 a(n-1) + 3 a(n-2)\). - INPUT: - n– nonnegative integer
 - OUTPUT: integer; function value - EXAMPLES: - sage: a = sloane.A015530;a Linear 2nd order recurrence, a(n) = 4 a(n-1) + 3 a(n-2). sage: a(0) 0 sage: a(1) 1 sage: a(2) 4 sage: a.offset 0 sage: a(8) 41008 sage: a.list(9) [0, 1, 4, 19, 88, 409, 1900, 8827, 41008] - >>> from sage.all import * >>> a = sloane.A015530;a Linear 2nd order recurrence, a(n) = 4 a(n-1) + 3 a(n-2). >>> a(Integer(0)) 0 >>> a(Integer(1)) 1 >>> a(Integer(2)) 4 >>> a.offset 0 >>> a(Integer(8)) 41008 >>> a.list(Integer(9)) [0, 1, 4, 19, 88, 409, 1900, 8827, 41008] - AUTHORS: - Jaap Spies (2007-01-19) 
 
- class sage.combinat.sloane_functions.A015531[source]¶
- Bases: - RecurrenceSequence2- Linear 2nd order recurrence, \(a(0)=0\), \(a(1)=1\) and \(a(n) = 4 a(n-1) + 5 a(n-2)\). - INPUT: - n– nonnegative integer
 - OUTPUT: integer; function value - EXAMPLES: - sage: a = sloane.A015531;a Linear 2nd order recurrence, a(n) = 4 a(n-1) + 5 a(n-2). sage: a(0) 0 sage: a(1) 1 sage: a(2) 4 sage: a.offset 0 sage: a(8) 65104 sage: a(60) 144560289664733924534327040115992228190104 sage: a.list(9) [0, 1, 4, 21, 104, 521, 2604, 13021, 65104] - >>> from sage.all import * >>> a = sloane.A015531;a Linear 2nd order recurrence, a(n) = 4 a(n-1) + 5 a(n-2). >>> a(Integer(0)) 0 >>> a(Integer(1)) 1 >>> a(Integer(2)) 4 >>> a.offset 0 >>> a(Integer(8)) 65104 >>> a(Integer(60)) 144560289664733924534327040115992228190104 >>> a.list(Integer(9)) [0, 1, 4, 21, 104, 521, 2604, 13021, 65104] - AUTHORS: - Jaap Spies (2007-01-19) 
 
- class sage.combinat.sloane_functions.A015551[source]¶
- Bases: - RecurrenceSequence2- Linear 2nd order recurrence, \(a(0)=0\), \(a(1)=1\) and \(a(n) = 6 a(n-1) + 5 a(n-2)\). - INPUT: - n– nonnegative integer
 - OUTPUT: integer; function value - EXAMPLES: - sage: a = sloane.A015551;a Linear 2nd order recurrence, a(n) = 6 a(n-1) + 5 a(n-2). sage: a(0) 0 sage: a(1) 1 sage: a(2) 6 sage: a.offset 0 sage: a(8) 570216 sage: a(60) 7110606606530059736761484557155863822531970573036 sage: a.list(9) [0, 1, 6, 41, 276, 1861, 12546, 84581, 570216] - >>> from sage.all import * >>> a = sloane.A015551;a Linear 2nd order recurrence, a(n) = 6 a(n-1) + 5 a(n-2). >>> a(Integer(0)) 0 >>> a(Integer(1)) 1 >>> a(Integer(2)) 6 >>> a.offset 0 >>> a(Integer(8)) 570216 >>> a(Integer(60)) 7110606606530059736761484557155863822531970573036 >>> a.list(Integer(9)) [0, 1, 6, 41, 276, 1861, 12546, 84581, 570216] - AUTHORS: - Jaap Spies (2007-01-19) 
 
- class sage.combinat.sloane_functions.A018252[source]¶
- Bases: - SloaneSequence- The nonprime numbers, starting with 1. - INPUT: - n– nonnegative integer
 - OUTPUT: integer; function value - EXAMPLES: - sage: a = sloane.A018252;a The nonprime numbers. sage: a(0) Traceback (most recent call last): ... ValueError: input n (=0) must be a positive integer sage: a(1) 1 sage: a(2) 4 sage: a(9) 15 sage: a.list(10) [1, 4, 6, 8, 9, 10, 12, 14, 15, 16] - >>> from sage.all import * >>> a = sloane.A018252;a The nonprime numbers. >>> a(Integer(0)) Traceback (most recent call last): ... ValueError: input n (=0) must be a positive integer >>> a(Integer(1)) 1 >>> a(Integer(2)) 4 >>> a(Integer(9)) 15 >>> a.list(Integer(10)) [1, 4, 6, 8, 9, 10, 12, 14, 15, 16] - AUTHORS: - Jaap Spies (2007-02-04) 
 
- class sage.combinat.sloane_functions.A020639[source]¶
- Bases: - SloaneSequence- Least prime dividing \(n\) with \(a(1)=1\). - INPUT: - n– nonnegative integer
 - OUTPUT: integer; function value - EXAMPLES: - sage: a = sloane.A020639;a Least prime dividing n (a(1)=1). sage: a(0) Traceback (most recent call last): ... ValueError: input n (=0) must be a positive integer sage: a(1) 1 sage: a(13) 13 sage: a.list(14) [1, 2, 3, 2, 5, 2, 7, 2, 3, 2, 11, 2, 13, 2] - >>> from sage.all import * >>> a = sloane.A020639;a Least prime dividing n (a(1)=1). >>> a(Integer(0)) Traceback (most recent call last): ... ValueError: input n (=0) must be a positive integer >>> a(Integer(1)) 1 >>> a(Integer(13)) 13 >>> a.list(Integer(14)) [1, 2, 3, 2, 5, 2, 7, 2, 3, 2, 11, 2, 13, 2] - AUTHORS: - Jaap Spies (2007-01-25) 
 
- class sage.combinat.sloane_functions.A046660(offset=1)[source]¶
- Bases: - SloaneSequence- Excess of \(n\) = number of prime divisors (with multiplicity) - number of prime divisors (without multiplicity). - \(\Omega(n) - \omega(n)\). - INPUT: - n– positive integer
 - OUTPUT: integer; function value - EXAMPLES: - sage: a = sloane.A046660; a Excess of n = Bigomega (with multiplicity) - omega (without multiplicity). sage: a(0) Traceback (most recent call last): ... ValueError: input n (=0) must be a positive integer sage: a(1) 0 sage: a(8) 2 sage: a(41) 0 sage: a(84792) 2 sage: a.list(12) [0, 0, 0, 1, 0, 0, 0, 2, 1, 0, 0, 1] - >>> from sage.all import * >>> a = sloane.A046660; a Excess of n = Bigomega (with multiplicity) - omega (without multiplicity). >>> a(Integer(0)) Traceback (most recent call last): ... ValueError: input n (=0) must be a positive integer >>> a(Integer(1)) 0 >>> a(Integer(8)) 2 >>> a(Integer(41)) 0 >>> a(Integer(84792)) 2 >>> a.list(Integer(12)) [0, 0, 0, 1, 0, 0, 0, 2, 1, 0, 0, 1] - AUTHORS: - Jaap Spies (2007-01-19) 
 
- class sage.combinat.sloane_functions.A049310[source]¶
- Bases: - SloaneSequence- Triangle of coefficients of Chebyshev’s \(S(n,x)\): \(U(n, \frac x 2)\) polynomials (exponents in increasing order). - INPUT: - n– nonnegative integer
 - OUTPUT: integer; function value - EXAMPLES: - sage: a = sloane.A049310;a Triangle of coefficients of Chebyshev's S(n,x) := U(n,x/2) polynomials (exponents in increasing order). sage: a(0) 1 sage: a(1) 0 sage: a(13) 0 sage: a.list(15) [1, 0, 1, -1, 0, 1, 0, -2, 0, 1, 1, 0, -3, 0, 1] sage: a(200) 0 sage: a.keyword ['sign', 'tabl', 'nice', 'easy', 'core', 'triangle'] - >>> from sage.all import * >>> a = sloane.A049310;a Triangle of coefficients of Chebyshev's S(n,x) := U(n,x/2) polynomials (exponents in increasing order). >>> a(Integer(0)) 1 >>> a(Integer(1)) 0 >>> a(Integer(13)) 0 >>> a.list(Integer(15)) [1, 0, 1, -1, 0, 1, 0, -2, 0, 1, 1, 0, -3, 0, 1] >>> a(Integer(200)) 0 >>> a.keyword ['sign', 'tabl', 'nice', 'easy', 'core', 'triangle'] - AUTHORS: - Jaap Spies (2007-01-31) 
 - keyword = ['sign', 'tabl', 'nice', 'easy', 'core', 'triangle']¶
 
- class sage.combinat.sloane_functions.A051959[source]¶
- Bases: - RecurrenceSequence- Linear second order recurrence. A051959. - INPUT: - n– nonnegative integer
 - OUTPUT: integer; function value - EXAMPLES: - sage: a = sloane.A051959; a Linear second order recurrence. A051959. sage: a(0) 1 sage: a(1) 10 sage: a(8) 9969 sage: a(41) 42834431872413650 sage: a.list(12) [1, 10, 36, 104, 273, 686, 1688, 4112, 9969, 24114, 58268, 140728] - >>> from sage.all import * >>> a = sloane.A051959; a Linear second order recurrence. A051959. >>> a(Integer(0)) 1 >>> a(Integer(1)) 10 >>> a(Integer(8)) 9969 >>> a(Integer(41)) 42834431872413650 >>> a.list(Integer(12)) [1, 10, 36, 104, 273, 686, 1688, 4112, 9969, 24114, 58268, 140728] - AUTHORS: - Jaap Spies (2007-01-19) 
 
- class sage.combinat.sloane_functions.A055790[source]¶
- Bases: - ExtremesOfPermanentsSequence2- \(a(n) = n*a(n-1) + (n-2)*a(n-2) [a(0) = 0, a(1) = 2]\). - With offset 1, permanent of (0,1)-matrix of size n X (n+d) with d=1 and n-1 zeros not on a line. This is a special case of Theorem 2.3 of Seok-Zun Song et al. Extremes of permanents of (0,1)-matrices, p. 201-202. - REFERENCES: - Seok-Zun Song et al., Extremes of permanents of (0,1)-matrices, Lin. Algebra and its Applic. 373 (2003), p. 197-210. 
 - INPUT: - n– nonnegative integer
 - OUTPUT: integer; function value - EXAMPLES: - sage: a = sloane.A055790;a a(n) = n*a(n-1) + (n-2)*a(n-2) [a(0) = 0, a(1) = 2]. sage: a(0) 0 sage: a(1) 2 sage: a(2) 4 sage: a.offset 0 sage: a(8) 165016 sage: a(22) 10356214297533070441564 sage: a.list(9) [0, 2, 4, 14, 64, 362, 2428, 18806, 165016] - >>> from sage.all import * >>> a = sloane.A055790;a a(n) = n*a(n-1) + (n-2)*a(n-2) [a(0) = 0, a(1) = 2]. >>> a(Integer(0)) 0 >>> a(Integer(1)) 2 >>> a(Integer(2)) 4 >>> a.offset 0 >>> a(Integer(8)) 165016 >>> a(Integer(22)) 10356214297533070441564 >>> a.list(Integer(9)) [0, 2, 4, 14, 64, 362, 2428, 18806, 165016] - AUTHORS: - Jaap Spies (2007-01-23) 
 
- class sage.combinat.sloane_functions.A061084[source]¶
- Bases: - SloaneSequence- Fibonacci-type sequence based on subtraction: \(a(0) = 1\), \(a(1) = 2\) and \(a(n) = a(n-2)-a(n-1)\). - INPUT: - n– nonnegative integer
 - OUTPUT: integer; function value - EXAMPLES: - sage: a = sloane.A061084; a Fibonacci-type sequence based on subtraction: a(0) = 1, a(1) = 2 and a(n) = a(n-2)-a(n-1). sage: a(0) 1 sage: a(1) 2 sage: a(8) -29 sage: a(22) -24476 sage: a.list(12) [1, 2, -1, 3, -4, 7, -11, 18, -29, 47, -76, 123] sage: a.keyword ['sign', 'easy', 'nice'] - >>> from sage.all import * >>> a = sloane.A061084; a Fibonacci-type sequence based on subtraction: a(0) = 1, a(1) = 2 and a(n) = a(n-2)-a(n-1). >>> a(Integer(0)) 1 >>> a(Integer(1)) 2 >>> a(Integer(8)) -29 >>> a(Integer(22)) -24476 >>> a.list(Integer(12)) [1, 2, -1, 3, -4, 7, -11, 18, -29, 47, -76, 123] >>> a.keyword ['sign', 'easy', 'nice'] - AUTHORS: - Jaap Spies (2007-01-18) 
 - keyword = ['sign', 'easy', 'nice']¶
 
- class sage.combinat.sloane_functions.A064553[source]¶
- Bases: - SloaneSequence- \(a(1) = 1\), \(a(prime(i)) = i + 1\) for \(i > 0\) and \(a(u \cdot v) = a(u) \cdot a(v)\) for \(u, v > 0\). - INPUT: - n– nonnegative integer
 - OUTPUT: integer; function value - EXAMPLES: - sage: a = sloane.A064553;a a(1) = 1, a(prime(i)) = i+1 for i > 0 and a(u*v) = a(u)*a(v) for u,v > 0 sage: a(0) Traceback (most recent call last): ... ValueError: input n (=0) must be a positive integer sage: a(1) 1 sage: a(2) 2 sage: a(9) 9 sage: a.list(16) [1, 2, 3, 4, 4, 6, 5, 8, 9, 8, 6, 12, 7, 10, 12, 16] - >>> from sage.all import * >>> a = sloane.A064553;a a(1) = 1, a(prime(i)) = i+1 for i > 0 and a(u*v) = a(u)*a(v) for u,v > 0 >>> a(Integer(0)) Traceback (most recent call last): ... ValueError: input n (=0) must be a positive integer >>> a(Integer(1)) 1 >>> a(Integer(2)) 2 >>> a(Integer(9)) 9 >>> a.list(Integer(16)) [1, 2, 3, 4, 4, 6, 5, 8, 9, 8, 6, 12, 7, 10, 12, 16] - AUTHORS: - Jaap Spies (2007-02-04) 
 
- class sage.combinat.sloane_functions.A079922(offset=1)[source]¶
- Bases: - SloaneSequence- Function returns solutions to the Dancing School problem with \(n\) girls and \(n+3\) boys. - The value is \(per(B)\), the permanent of the (0,1)-matrix \(B\) of size \(n \times n+3\) with \(b(i,j)=1\) if and only if \(i \le j \le i+n\). - REFERENCES: - Jaap Spies, Nieuw Archief voor Wiskunde, 5/7 nr 4, December 2006 
 - INPUT: - n– positive integer
 - OUTPUT: integer; function value - EXAMPLES: - sage: a = sloane.A079922; a Solutions to the Dancing School problem with n girls and n+3 boys sage: a.offset 1 sage: a(1) 4 sage: a(8) 2227 sage: a.list(8) [4, 13, 36, 90, 212, 478, 1044, 2227] - >>> from sage.all import * >>> a = sloane.A079922; a Solutions to the Dancing School problem with n girls and n+3 boys >>> a.offset 1 >>> a(Integer(1)) 4 >>> a(Integer(8)) 2227 >>> a.list(Integer(8)) [4, 13, 36, 90, 212, 478, 1044, 2227] - Compare: Searching Sloane’s online database… Solution to the Dancing School Problem with n girls and n+3 boys: f(n,3). [4, 13, 36, 90, 212, 478, 1044, 2227] - sage: a(-1) Traceback (most recent call last): ... ValueError: input n (=-1) must be a positive integer - >>> from sage.all import * >>> a(-Integer(1)) Traceback (most recent call last): ... ValueError: input n (=-1) must be a positive integer - AUTHORS: - Jaap Spies (2007-01-14) 
 
- class sage.combinat.sloane_functions.A079923(offset=1)[source]¶
- Bases: - SloaneSequence- Function returns solutions to the Dancing School problem with \(n\) girls and \(n+4\) boys. - The value is \(per(B)\), the permanent of the (0,1)-matrix \(B\) of size \(n \times n+3\) with \(b(i,j)=1\) if and only if \(i \le j \le i+n\). - REFERENCES: - Jaap Spies, Nieuw Archief voor Wiskunde, 5/7 nr 4, December 2006 
 - INPUT: - n– positive integer
 - OUTPUT: integer; function value - EXAMPLES: - sage: a = sloane.A079923; a Solutions to the Dancing School problem with n girls and n+4 boys sage: a.offset 1 sage: a(1) 5 sage: a(8) 15458 sage: a.list(8) [5, 21, 76, 246, 738, 2108, 5794, 15458] - >>> from sage.all import * >>> a = sloane.A079923; a Solutions to the Dancing School problem with n girls and n+4 boys >>> a.offset 1 >>> a(Integer(1)) 5 >>> a(Integer(8)) 15458 >>> a.list(Integer(8)) [5, 21, 76, 246, 738, 2108, 5794, 15458] - Compare: Searching Sloane’s online database… Solution to the Dancing School Problem with n girls and n+4 boys: f(n,4). [5, 21, 76, 246, 738, 2108, 5794, 15458] - sage: a(0) Traceback (most recent call last): ... ValueError: input n (=0) must be a positive integer - >>> from sage.all import * >>> a(Integer(0)) Traceback (most recent call last): ... ValueError: input n (=0) must be a positive integer - AUTHORS: - Jaap Spies (2007-01-17) 
 
- class sage.combinat.sloane_functions.A082411[source]¶
- Bases: - RecurrenceSequence2- Second-order linear recurrence sequence with \(a(n) = a(n-1) + a(n-2)\). - \(a(0) = 407389224418\), \(a(1) = 76343678551\). This is the second-order linear recurrence sequence with \(a(0)\) and \(a(1)\) co-prime, that R. L. Graham in 1964 stated did not contain any primes. - INPUT: - n– nonnegative integer
 - OUTPUT: integer; function value - EXAMPLES: - sage: a = sloane.A082411;a Second-order linear recurrence sequence with a(n) = a(n-1) + a(n-2). sage: a(1) 76343678551 sage: a(2) 483732902969 sage: a(3) 560076581520 sage: a(20) 2219759332689173 sage: a.list(4) [407389224418, 76343678551, 483732902969, 560076581520] - >>> from sage.all import * >>> a = sloane.A082411;a Second-order linear recurrence sequence with a(n) = a(n-1) + a(n-2). >>> a(Integer(1)) 76343678551 >>> a(Integer(2)) 483732902969 >>> a(Integer(3)) 560076581520 >>> a(Integer(20)) 2219759332689173 >>> a.list(Integer(4)) [407389224418, 76343678551, 483732902969, 560076581520] - AUTHORS: - Jaap Spies (2007-01-23) 
 
- class sage.combinat.sloane_functions.A083103[source]¶
- Bases: - RecurrenceSequence2- Second-order linear recurrence sequence with \(a(n) = a(n-1) + a(n-2)\). - \(a(0) = 1786772701928802632268715130455793\), \(a(1) = 1059683225053915111058165141686995\). This is the second-order linear recurrence sequence with \(a(0)\) and \(a(1)\) co- prime, that R. L. Graham in 1964 stated did not contain any primes. It has not been verified. Graham made a mistake in the calculation that was corrected by D. E. Knuth in 1990. - INPUT: - n– nonnegative integer
 - OUTPUT: integer; function value - EXAMPLES: - sage: a = sloane.A083103;a Second-order linear recurrence sequence with a(n) = a(n-1) + a(n-2). sage: a(1) 1059683225053915111058165141686995 sage: a(2) 2846455926982717743326880272142788 sage: a(3) 3906139152036632854385045413829783 sage: a.offset 0 sage: a(8) 45481392851206651551714764671352204 sage: a(20) 14639253684254059531823985143948191708 sage: a.list(4) [1786772701928802632268715130455793, 1059683225053915111058165141686995, 2846455926982717743326880272142788, 3906139152036632854385045413829783] - >>> from sage.all import * >>> a = sloane.A083103;a Second-order linear recurrence sequence with a(n) = a(n-1) + a(n-2). >>> a(Integer(1)) 1059683225053915111058165141686995 >>> a(Integer(2)) 2846455926982717743326880272142788 >>> a(Integer(3)) 3906139152036632854385045413829783 >>> a.offset 0 >>> a(Integer(8)) 45481392851206651551714764671352204 >>> a(Integer(20)) 14639253684254059531823985143948191708 >>> a.list(Integer(4)) [1786772701928802632268715130455793, 1059683225053915111058165141686995, 2846455926982717743326880272142788, 3906139152036632854385045413829783] - AUTHORS: - Jaap Spies (2007-01-23) 
 
- class sage.combinat.sloane_functions.A083104[source]¶
- Bases: - RecurrenceSequence2- Second-order linear recurrence sequence with \(a(n) = a(n-1) + a(n-2)\). - \(a(0) = 331635635998274737472200656430763\), \(a(1) = 1510028911088401971189590305498785\). This is the second-order linear recurrence sequence with \(a(0)\) and \(a(1)\) co-prime. It was found by Ronald Graham in 1990. - INPUT: - n– nonnegative integer
 - OUTPUT: integer; function value - EXAMPLES: - sage: a = sloane.A083104;a Second-order linear recurrence sequence with a(n) = a(n-1) + a(n-2). sage: a(3) 3351693458175078679851381267428333 sage: a.offset 0 sage: a(8) 36021870400834012982120004949074404 sage: a(20) 11601914177621826012468849361236300628 - >>> from sage.all import * >>> a = sloane.A083104;a Second-order linear recurrence sequence with a(n) = a(n-1) + a(n-2). >>> a(Integer(3)) 3351693458175078679851381267428333 >>> a.offset 0 >>> a(Integer(8)) 36021870400834012982120004949074404 >>> a(Integer(20)) 11601914177621826012468849361236300628 - AUTHORS: - Jaap Spies (2007-01-23) 
 
- class sage.combinat.sloane_functions.A083105[source]¶
- Bases: - RecurrenceSequence2- Second-order linear recurrence sequence with \(a(n) = a(n-1) + a(n-2)\). - \(a(0) = 62638280004239857\), \(a(1) = 49463435743205655\). This is the second-order linear recurrence sequence with \(a(0)\) and \(a(1)\) co-prime. It was found by Donald Knuth in 1990. - INPUT: - n– nonnegative integer
 - OUTPUT: integer; function value - EXAMPLES: - sage: a = sloane.A083105;a Second-order linear recurrence sequence with a(n) = a(n-1) + a(n-2). sage: a(1) 49463435743205655 sage: a(2) 112101715747445512 sage: a(3) 161565151490651167 sage: a.offset 0 sage: a(8) 1853029790662436896 sage: a(20) 596510791500513098192 sage: a.list(4) [62638280004239857, 49463435743205655, 112101715747445512, 161565151490651167] - >>> from sage.all import * >>> a = sloane.A083105;a Second-order linear recurrence sequence with a(n) = a(n-1) + a(n-2). >>> a(Integer(1)) 49463435743205655 >>> a(Integer(2)) 112101715747445512 >>> a(Integer(3)) 161565151490651167 >>> a.offset 0 >>> a(Integer(8)) 1853029790662436896 >>> a(Integer(20)) 596510791500513098192 >>> a.list(Integer(4)) [62638280004239857, 49463435743205655, 112101715747445512, 161565151490651167] - AUTHORS: - Jaap Spies (2007-01-23) 
 
- class sage.combinat.sloane_functions.A083216[source]¶
- Bases: - RecurrenceSequence2- Second-order linear recurrence sequence with \(a(n) = a(n-1) + a(n-2)\). - \(a(0) = 20615674205555510\), \(a(1) = 3794765361567513\). This is a second-order linear recurrence sequence with \(a(0)\) and \(a(1)\) co-prime that does not contain any primes. It was found by Herbert Wilf in 1990. - INPUT: - n– nonnegative integer
 - OUTPUT: integer; function value - EXAMPLES: - sage: a = sloane.A083216; a Second-order linear recurrence sequence with a(n) = a(n-1) + a(n-2). sage: a(0) 20615674205555510 sage: a(1) 3794765361567513 sage: a(8) 347693837265139403 sage: a(41) 2738025383211084205003383 sage: a.list(4) [20615674205555510, 3794765361567513, 24410439567123023, 28205204928690536] - >>> from sage.all import * >>> a = sloane.A083216; a Second-order linear recurrence sequence with a(n) = a(n-1) + a(n-2). >>> a(Integer(0)) 20615674205555510 >>> a(Integer(1)) 3794765361567513 >>> a(Integer(8)) 347693837265139403 >>> a(Integer(41)) 2738025383211084205003383 >>> a.list(Integer(4)) [20615674205555510, 3794765361567513, 24410439567123023, 28205204928690536] - AUTHORS: - Jaap Spies (2007-01-19) 
 
- class sage.combinat.sloane_functions.A090010[source]¶
- Bases: - ExtremesOfPermanentsSequence2- Permanent of (0,1)-matrix of size \(n \times (n+d)\) with \(d=6\) and \(n\) zeros not on a line. - ` a(n) = (n+5)*a(n-1) + (n-1)*a(n-2), a(1)=6, a(2)=43`. - This is a special case of Theorem 2.3 of Seok-Zun Song et al. Extremes of permanents of (0,1)-matrices, p. 201-202. - REFERENCES: - Seok-Zun Song et al., Extremes of permanents of (0,1)-matrices, Lin. Algebra and its Applic. 373 (2003), p. 197-210. 
 - INPUT: - n– nonnegative integer
 - OUTPUT: integer; function value - EXAMPLES: - sage: a = sloane.A090010;a Permanent of (0,1)-matrix of size n X (n+d) with d=6 and n zeros not on a line. sage: a(0) Traceback (most recent call last): ... ValueError: input n (=0) must be a positive integer sage: a(1) 6 sage: a(2) 43 sage: a.offset 1 sage: a(8) 67741129 sage: a(22) 192416593029158989003270143 sage: a.list(9) [6, 43, 356, 3333, 34754, 398959, 4996032, 67741129, 988344062] - >>> from sage.all import * >>> a = sloane.A090010;a Permanent of (0,1)-matrix of size n X (n+d) with d=6 and n zeros not on a line. >>> a(Integer(0)) Traceback (most recent call last): ... ValueError: input n (=0) must be a positive integer >>> a(Integer(1)) 6 >>> a(Integer(2)) 43 >>> a.offset 1 >>> a(Integer(8)) 67741129 >>> a(Integer(22)) 192416593029158989003270143 >>> a.list(Integer(9)) [6, 43, 356, 3333, 34754, 398959, 4996032, 67741129, 988344062] - AUTHORS: - Jaap Spies (2007-01-23) 
 
- class sage.combinat.sloane_functions.A090012[source]¶
- Bases: - SloaneSequence- Permanent of (0,1)-matrix of size \(n \times (n+d)\) with \(d=2\) and \(n-1\) zeros not on a line. - \(a(n) = (n+1)*a(n-1) + (n-2)*a(n-2)\), \(a(1)=3\) and \(a(2)=9\) - This is a special case of Theorem 2.3 of Seok-Zun Song et al. Extremes of permanents of (0,1)-matrices, p. 201-202. - REFERENCES: - Seok-Zun Song et al., Extremes of permanents of (0,1)-matrices, Lin. Algebra and its Applic. 373 (2003), p. 197-210. 
 - INPUT: - n– nonnegative integer
 - OUTPUT: integer; function value - EXAMPLES: - sage: a = sloane.A090012;a Permanent of (0,1)-matrix of size n X (n+d) with d=2 and n-1 zeros not on a line. sage: a(0) Traceback (most recent call last): ... ValueError: input n (=0) must be a positive integer sage: a(1) 3 sage: a(2) 9 sage: a.offset 1 sage: a(8) 890901 sage: a(22) 129020386652297208795129 sage: a.list(9) [3, 9, 39, 213, 1395, 10617, 91911, 890901, 9552387] - >>> from sage.all import * >>> a = sloane.A090012;a Permanent of (0,1)-matrix of size n X (n+d) with d=2 and n-1 zeros not on a line. >>> a(Integer(0)) Traceback (most recent call last): ... ValueError: input n (=0) must be a positive integer >>> a(Integer(1)) 3 >>> a(Integer(2)) 9 >>> a.offset 1 >>> a(Integer(8)) 890901 >>> a(Integer(22)) 129020386652297208795129 >>> a.list(Integer(9)) [3, 9, 39, 213, 1395, 10617, 91911, 890901, 9552387] - AUTHORS: - Jaap Spies (2007-01-23) 
 
- class sage.combinat.sloane_functions.A090013[source]¶
- Bases: - SloaneSequence- Permanent of (0,1)-matrix of size \(n \times (n+d)\) with \(d=3\) and \(n-1\) zeros not on a line. - \(a(n) = (n+1)*a(n-1) + (n-2)*a(n-2) [a(1)=4, a(2)=16]\) - This is a special case of Theorem 2.3 of Seok-Zun Song et al. Extremes of permanents of (0,1)-matrices, p. 201-202. - REFERENCES: - Seok-Zun Song et al., Extremes of permanents of (0,1)-matrices, Lin. Algebra and its Applic. 373 (2003), p. 197-210. 
 - INPUT: - n– nonnegative integer
 - OUTPUT: integer; function value - EXAMPLES: - sage: a = sloane.A090013;a Permanent of (0,1)-matrix of size n X (n+d) with d=3 and n-1 zeros not on a line. sage: a(0) Traceback (most recent call last): ... ValueError: input n (=0) must be a positive integer sage: a(1) 4 sage: a(2) 16 sage: a.offset 1 sage: a(8) 3481096 sage: a(22) 1112998577171142607670336 sage: a.list(9) [4, 16, 84, 536, 4004, 34176, 327604, 3481096, 40585284] - >>> from sage.all import * >>> a = sloane.A090013;a Permanent of (0,1)-matrix of size n X (n+d) with d=3 and n-1 zeros not on a line. >>> a(Integer(0)) Traceback (most recent call last): ... ValueError: input n (=0) must be a positive integer >>> a(Integer(1)) 4 >>> a(Integer(2)) 16 >>> a.offset 1 >>> a(Integer(8)) 3481096 >>> a(Integer(22)) 1112998577171142607670336 >>> a.list(Integer(9)) [4, 16, 84, 536, 4004, 34176, 327604, 3481096, 40585284] - AUTHORS: - Jaap Spies (2007-01-23) 
 
- class sage.combinat.sloane_functions.A090014[source]¶
- Bases: - SloaneSequence- Permanent of (0,1)-matrix of size \(n \times (n+d)\) with \(d=4\) and \(n-1\) zeros not on a line. - \(a(n) = (n+1)*a(n-1) + (n-2)*a(n-2) [a(1)=5, a(2)=25]\) - This is a special case of Theorem 2.3 of Seok-Zun Song et al. Extremes of permanents of (0,1)-matrices, p. 201-202. - REFERENCES: - Seok-Zun Song et al., Extremes of permanents of (0,1)-matrices, Lin. Algebra and its Applic. 373 (2003), p. 197-210. 
 - INPUT: - n– nonnegative integer
 - OUTPUT: integer; function value - EXAMPLES: - sage: a = sloane.A090014;a Permanent of (0,1)-matrix of size n X (n+d) with d=4 and n-1 zeros not on a line. sage: a(0) Traceback (most recent call last): ... ValueError: input n (=0) must be a positive integer sage: a(1) 5 sage: a(2) 25 sage: a.offset 1 sage: a(8) 11016595 sage: a(22) 7469733600354446865509725 sage: a.list(9) [5, 25, 155, 1135, 9545, 90445, 952175, 11016595, 138864365] - >>> from sage.all import * >>> a = sloane.A090014;a Permanent of (0,1)-matrix of size n X (n+d) with d=4 and n-1 zeros not on a line. >>> a(Integer(0)) Traceback (most recent call last): ... ValueError: input n (=0) must be a positive integer >>> a(Integer(1)) 5 >>> a(Integer(2)) 25 >>> a.offset 1 >>> a(Integer(8)) 11016595 >>> a(Integer(22)) 7469733600354446865509725 >>> a.list(Integer(9)) [5, 25, 155, 1135, 9545, 90445, 952175, 11016595, 138864365] - AUTHORS: - Jaap Spies (2007-01-23) 
 
- class sage.combinat.sloane_functions.A090015[source]¶
- Bases: - SloaneSequence- Permanent of (0,1)-matrix of size \(n \times (n+d)\) with \(d=5\) and \(n-1\) zeros not on a line. - \(a(n) = (n+1)*a(n-1) + (n-2)*a(n-2) [a(1)=6, a(2)=36]\) - This is a special case of Theorem 2.3 of Seok-Zun Song et al. Extremes of permanents of (0,1)-matrices, p. 201-202. - REFERENCES: - Seok-Zun Song et al., Extremes of permanents of (0,1)-matrices, Lin. Algebra and its Applic. 373 (2003), p. 197-210. 
 - INPUT: - n– nonnegative integer
 - OUTPUT: integer; function value - EXAMPLES: - sage: a = sloane.A090015;a Permanent of (0,1)-matrix of size n X (n+d) with d=3 and n-1 zeros not on a line. sage: a(0) Traceback (most recent call last): ... ValueError: input n (=0) must be a positive integer sage: a(1) 6 sage: a(2) 36 sage: a.offset 1 sage: a(8) 29976192 sage: a(22) 41552258517692116794936876 sage: a.list(9) [6, 36, 258, 2136, 19998, 208524, 2393754, 29976192, 406446774] - >>> from sage.all import * >>> a = sloane.A090015;a Permanent of (0,1)-matrix of size n X (n+d) with d=3 and n-1 zeros not on a line. >>> a(Integer(0)) Traceback (most recent call last): ... ValueError: input n (=0) must be a positive integer >>> a(Integer(1)) 6 >>> a(Integer(2)) 36 >>> a.offset 1 >>> a(Integer(8)) 29976192 >>> a(Integer(22)) 41552258517692116794936876 >>> a.list(Integer(9)) [6, 36, 258, 2136, 19998, 208524, 2393754, 29976192, 406446774] - AUTHORS: - Jaap Spies (2007-01-23) 
 
- class sage.combinat.sloane_functions.A090016[source]¶
- Bases: - SloaneSequence- Permanent of (0,1)-matrix of size \(n \times (n+d)\) with \(d=6\) and \(n-1\) zeros not on a line. - \(a(n) = (n+1)*a(n-1) + (n-2)*a(n-2) [a(1)=7, a(2)=49]\) - \(A090016 a(n) = A090010(n-1) + A090010(n), a(1)=7\) - This is a special case of Theorem 2.3 of Seok-Zun Song et al. Extremes of permanents of (0,1)-matrices, p. 201-202. - REFERENCES: - Seok-Zun Song et al., Extremes of permanents of (0,1)-matrices, Lin. Algebra and its Applic. 373 (2003), p. 197-210. 
 - INPUT: - n– nonnegative integer
 - OUTPUT: integer; function value - EXAMPLES: - sage: a = sloane.A090016;a Permanent of (0,1)-matrix of size n X (n+d) with d=6 and n-1 zeros not on a line. sage: a(0) Traceback (most recent call last): ... ValueError: input n (=0) must be a positive integer sage: a(1) 7 sage: a(2) 49 sage: a.offset 1 sage: a(8) 72737161 sage: a(22) 199341969448774341802426289 sage: a.list(9) [7, 49, 399, 3689, 38087, 433713, 5394991, 72737161, 1056085191] - >>> from sage.all import * >>> a = sloane.A090016;a Permanent of (0,1)-matrix of size n X (n+d) with d=6 and n-1 zeros not on a line. >>> a(Integer(0)) Traceback (most recent call last): ... ValueError: input n (=0) must be a positive integer >>> a(Integer(1)) 7 >>> a(Integer(2)) 49 >>> a.offset 1 >>> a(Integer(8)) 72737161 >>> a(Integer(22)) 199341969448774341802426289 >>> a.list(Integer(9)) [7, 49, 399, 3689, 38087, 433713, 5394991, 72737161, 1056085191] - AUTHORS: - Jaap Spies (2007-01-23) 
 
- class sage.combinat.sloane_functions.A109814[source]¶
- Bases: - SloaneSequence- The \(n\) th term of the sequence \(a(n)\) is the largest \(k\) such that \(n\) can be written as sum of \(k\) consecutive integers. - By definition, \(n\) is the sum of at most \(a(n)\) consecutive positive integers. Suppose \(n\) is to be written as sum of \(k\) consecutive integers starting with \(m\), then \(2n = k(2m + k - 1)\). Only one of the factors is odd. For each odd divisor \(d\) of \(n\) there is a unique corresponding \(k = min(d,2n/d)\). \(a(n)\) can be alternatively defined as the largest among those \(k\) . - See also - INPUT: - n– nonnegative integer
 - OUTPUT: integer; function value - EXAMPLES: - sage: a = sloane.A109814; a a(n) is the largest k such that n can be written as sum of k consecutive positive integers. sage: a(0) Traceback (most recent call last): ... ValueError: input n (=0) must be a positive integer sage: a(2) 1 sage: a.list(9) [1, 1, 2, 1, 2, 3, 2, 1, 3] - >>> from sage.all import * >>> a = sloane.A109814; a a(n) is the largest k such that n can be written as sum of k consecutive positive integers. >>> a(Integer(0)) Traceback (most recent call last): ... ValueError: input n (=0) must be a positive integer >>> a(Integer(2)) 1 >>> a.list(Integer(9)) [1, 1, 2, 1, 2, 3, 2, 1, 3] - AUTHORS: - Jaap Spies (2007-01-13) 
 
- class sage.combinat.sloane_functions.A111774[source]¶
- Bases: - SloaneSequence- Sequence of numbers of the third kind, i.e., numbers that can be written as a sum of at least three consecutive positive integers. - Odd primes can only be written as a sum of two consecutive integers. Powers of 2 do not have a representation as a sum of \(k\) consecutive integers (other than the trivial \(n = n\) for \(k = 1\)). - See: http://www.jaapspies.nl/mathfiles/problem2005-2C.pdf - INPUT: - n– nonnegative integer
 - OUTPUT: integer; function value - EXAMPLES: - sage: a = sloane.A111774; a Numbers that can be written as a sum of at least three consecutive positive integers. sage: a(1) 6 sage: a(0) Traceback (most recent call last): ... ValueError: input n (=0) must be a positive integer sage: a(100) 141 sage: a(156) 209 sage: a(302) 386 sage: a.list(12) [6, 9, 10, 12, 14, 15, 18, 20, 21, 22, 24, 25] sage: a(1/3) Traceback (most recent call last): ... TypeError: input must be an int or Integer - >>> from sage.all import * >>> a = sloane.A111774; a Numbers that can be written as a sum of at least three consecutive positive integers. >>> a(Integer(1)) 6 >>> a(Integer(0)) Traceback (most recent call last): ... ValueError: input n (=0) must be a positive integer >>> a(Integer(100)) 141 >>> a(Integer(156)) 209 >>> a(Integer(302)) 386 >>> a.list(Integer(12)) [6, 9, 10, 12, 14, 15, 18, 20, 21, 22, 24, 25] >>> a(Integer(1)/Integer(3)) Traceback (most recent call last): ... TypeError: input must be an int or Integer - AUTHORS: - Jaap Spies (2007-01-13) 
 - is_number_of_the_third_kind(n)[source]¶
- Return - Trueif and only if \(n\) is a number of the third kind.- A number is of the third kind if it can be written as a sum of at least three consecutive positive integers. Odd primes can only be written as a sum of two consecutive integers. Powers of 2 do not have a representation as a sum of \(k\) consecutive integers (other than the trivial \(n = n\) for \(k = 1\)). - See: http://www.jaapspies.nl/mathfiles/problem2005-2C.pdf - INPUT: - n– positive integer
 - OUTPUT: - Trueif \(n\) is not prime and not a power of 2- EXAMPLES: - sage: a = sloane.A111774 sage: a.is_number_of_the_third_kind(6) True sage: a.is_number_of_the_third_kind(100) True sage: a.is_number_of_the_third_kind(16) False sage: a.is_number_of_the_third_kind(97) False - >>> from sage.all import * >>> a = sloane.A111774 >>> a.is_number_of_the_third_kind(Integer(6)) True >>> a.is_number_of_the_third_kind(Integer(100)) True >>> a.is_number_of_the_third_kind(Integer(16)) False >>> a.is_number_of_the_third_kind(Integer(97)) False - AUTHORS: - Jaap Spies (2006-12-09) 
 
 
- class sage.combinat.sloane_functions.A111775[source]¶
- Bases: - SloaneSequence- Number of ways \(n\) can be written as a sum of at least three consecutive integers. - Powers of 2 and (odd) primes can not be written as a sum of at least three consecutive integers. \(a(n)\) strongly depends on the number of odd divisors of \(n\) (A001227): Suppose \(n\) is to be written as sum of \(k\) consecutive integers starting with \(m\), then \(2n = k(2m + k - 1)\). Only one of the factors is odd. For each odd divisor of \(n\) there is a unique corresponding \(k\), \(k=1\) and \(k=2\) must be excluded. - See: http://www.jaapspies.nl/mathfiles/problem2005-2C.pdf - INPUT: - n– nonnegative integer
 - OUTPUT: integer; function value - EXAMPLES: - sage: a = sloane.A111775; a Number of ways n can be written as a sum of at least three consecutive integers. - >>> from sage.all import * >>> a = sloane.A111775; a Number of ways n can be written as a sum of at least three consecutive integers. - sage: a(1) 0 sage: a(0) 0 - >>> from sage.all import * >>> a(Integer(1)) 0 >>> a(Integer(0)) 0 - We have a(15)=2 because 15 = 4+5+6 and 15 = 1+2+3+4+5. The number of odd divisors of 15 is 4. - sage: a(15) 2 - >>> from sage.all import * >>> a(Integer(15)) 2 - sage: a(100) 2 sage: a(256) 0 sage: a(29) 0 sage: a.list(20) [0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 1, 0, 1, 2, 0, 0, 2, 0] sage: a(1/3) Traceback (most recent call last): ... TypeError: input must be an int or Integer - >>> from sage.all import * >>> a(Integer(100)) 2 >>> a(Integer(256)) 0 >>> a(Integer(29)) 0 >>> a.list(Integer(20)) [0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 1, 0, 1, 2, 0, 0, 2, 0] >>> a(Integer(1)/Integer(3)) Traceback (most recent call last): ... TypeError: input must be an int or Integer - AUTHORS: - Jaap Spies (2006-12-09) 
 
- class sage.combinat.sloane_functions.A111787[source]¶
- Bases: - SloaneSequence- This function returns the \(n\)-th number of Sloane’s sequence A111787 - \(a(n)=0\) if \(n\) is an odd prime or a power of 2. For numbers of the third kind (see A111774) we proceed as follows: suppose \(n\) is to be written as sum of \(k\) consecutive integers starting with \(m\), then \(2n = k(2m + k - 1)\). Let \(p\) be the smallest odd prime divisor of \(n\) then \(a(n) = min(p,2n/p)\). - See: http://www.jaapspies.nl/mathfiles/problem2005-2C.pdf - INPUT: - n– nonnegative integer
 - OUTPUT: integer; function value - EXAMPLES: - sage: a = sloane.A111787; a a(n) is the least k >= 3 such that n can be written as sum of k consecutive integers. a(n)=0 if such a k does not exist. sage: a.offset 1 sage: a(1) 0 sage: a(0) Traceback (most recent call last): ... ValueError: input n (=0) must be a positive integer sage: a(100) 5 sage: a(256) 0 sage: a(29) 0 sage: a.list(20) [0, 0, 0, 0, 0, 3, 0, 0, 3, 4, 0, 3, 0, 4, 3, 0, 0, 3, 0, 5] sage: a(-1) Traceback (most recent call last): ... ValueError: input n (=-1) must be a positive integer - >>> from sage.all import * >>> a = sloane.A111787; a a(n) is the least k >= 3 such that n can be written as sum of k consecutive integers. a(n)=0 if such a k does not exist. >>> a.offset 1 >>> a(Integer(1)) 0 >>> a(Integer(0)) Traceback (most recent call last): ... ValueError: input n (=0) must be a positive integer >>> a(Integer(100)) 5 >>> a(Integer(256)) 0 >>> a(Integer(29)) 0 >>> a.list(Integer(20)) [0, 0, 0, 0, 0, 3, 0, 0, 3, 4, 0, 3, 0, 4, 3, 0, 0, 3, 0, 5] >>> a(-Integer(1)) Traceback (most recent call last): ... ValueError: input n (=-1) must be a positive integer - AUTHORS: - Jaap Spies (2007-01-14) 
 
- class sage.combinat.sloane_functions.ExponentialNumbers(a)[source]¶
- Bases: - SloaneSequence- A sequence of Exponential numbers. - EXAMPLES: - sage: from sage.combinat.sloane_functions import ExponentialNumbers sage: ExponentialNumbers(0) Sequence of Exponential numbers around 0 - >>> from sage.all import * >>> from sage.combinat.sloane_functions import ExponentialNumbers >>> ExponentialNumbers(Integer(0)) Sequence of Exponential numbers around 0 
- class sage.combinat.sloane_functions.ExtremesOfPermanentsSequence(offset=1)[source]¶
- Bases: - SloaneSequence
- class sage.combinat.sloane_functions.ExtremesOfPermanentsSequence2(offset=1)[source]¶
- Bases: - ExtremesOfPermanentsSequence- gen(a0, a1, d)[source]¶
- EXAMPLES: - sage: from sage.combinat.sloane_functions import ExtremesOfPermanentsSequence2 sage: e = ExtremesOfPermanentsSequence2() sage: it = e.gen(6,43,6) sage: [next(it) for i in range(5)] [6, 43, 307, 2542, 23799] - >>> from sage.all import * >>> from sage.combinat.sloane_functions import ExtremesOfPermanentsSequence2 >>> e = ExtremesOfPermanentsSequence2() >>> it = e.gen(Integer(6),Integer(43),Integer(6)) >>> [next(it) for i in range(Integer(5))] [6, 43, 307, 2542, 23799] 
 
- class sage.combinat.sloane_functions.RecurrenceSequence(offset=1)[source]¶
- Bases: - SloaneSequence
- class sage.combinat.sloane_functions.RecurrenceSequence2(offset=1)[source]¶
- Bases: - SloaneSequence
- class sage.combinat.sloane_functions.Sloane[source]¶
- Bases: - SageObject- A collection of Sloane generating functions. - This class inspects sage.combinat.sloane_functions, accumulating all the SloaneSequence classes starting with ‘A’. These are listed for tab completion, but not instantiated until requested. - EXAMPLES: - Ensure we have lots of entries: - sage: len(sloane.__dir__()) > 100 True - >>> from sage.all import * >>> len(sloane.__dir__()) > Integer(100) True - Ensure none are being incorrectly returned: - sage: [ None for n in sloane.__dir__() if not n.startswith('A') ] [] - >>> from sage.all import * >>> [ None for n in sloane.__dir__() if not n.startswith('A') ] [] - Ensure we can access dynamic constructions and cache correctly: - sage: s = sloane.A000587 sage: s is sloane.A000587 True - >>> from sage.all import * >>> s = sloane.A000587 >>> s is sloane.A000587 True - Ensure that we can access other functions in parent classes: - sage: sloane.__class__ <class 'sage.combinat.sloane_functions.Sloane'> - >>> from sage.all import * >>> sloane.__class__ <class 'sage.combinat.sloane_functions.Sloane'> - AUTHORS: - Nick Alexander 
 
- class sage.combinat.sloane_functions.SloaneSequence(offset=1)[source]¶
- Bases: - SageObject- Base class for a Sloane integer sequence. 
- sage.combinat.sloane_functions.perm_mh(m, h)[source]¶
- This functions calculates \(f(g,h)\) from Sloane’s sequences A079908-A079928 - INPUT: - m– positive integer
- h– nonnegative integer
 - OUTPUT: permanent of the \(m \times (m+h)\) matrix, etc. - EXAMPLES: - sage: from sage.combinat.sloane_functions import perm_mh sage: perm_mh(3,3) 36 sage: perm_mh(3,4) 76 - >>> from sage.all import * >>> from sage.combinat.sloane_functions import perm_mh >>> perm_mh(Integer(3),Integer(3)) 36 >>> perm_mh(Integer(3),Integer(4)) 76 - AUTHORS: - Jaap Spies (2006) 
 
- sage.combinat.sloane_functions.recur_gen2(a0, a1, a2, a3)[source]¶
- Homogeneous general second-order linear recurrence generator with fixed coefficients. - a(0) = a0, a(1) = a1, a(n) = a2*a(n-1) + a3*a(n-2) - EXAMPLES: - sage: from sage.combinat.sloane_functions import recur_gen2 sage: it = recur_gen2(1,1,1,1) sage: [next(it) for i in range(10)] [1, 1, 2, 3, 5, 8, 13, 21, 34, 55] - >>> from sage.all import * >>> from sage.combinat.sloane_functions import recur_gen2 >>> it = recur_gen2(Integer(1),Integer(1),Integer(1),Integer(1)) >>> [next(it) for i in range(Integer(10))] [1, 1, 2, 3, 5, 8, 13, 21, 34, 55] 
- sage.combinat.sloane_functions.recur_gen2b(a0, a1, a2, a3, b)[source]¶
- Inhomogeneous second-order linear recurrence generator with fixed coefficients and \(b = f(n)\) - \(a(0) = a0\), \(a(1) = a1\), \(a(n) = a2*a(n-1) + a3*a(n-2) +f(n)\). - EXAMPLES: - sage: from sage.combinat.sloane_functions import recur_gen2b sage: it = recur_gen2b(1,1,1,1, lambda n: 0) sage: [next(it) for i in range(10)] [1, 1, 2, 3, 5, 8, 13, 21, 34, 55] - >>> from sage.all import * >>> from sage.combinat.sloane_functions import recur_gen2b >>> it = recur_gen2b(Integer(1),Integer(1),Integer(1),Integer(1), lambda n: Integer(0)) >>> [next(it) for i in range(Integer(10))] [1, 1, 2, 3, 5, 8, 13, 21, 34, 55] 
- sage.combinat.sloane_functions.recur_gen3(a0, a1, a2, a3, a4, a5)[source]¶
- Homogeneous general third-order linear recurrence generator with fixed coefficients - a(0) = a0, a(1) = a1, a(2) = a2, a(n) = a3*a(n-1) + a4*a(n-2) + a5*a(n-3) - EXAMPLES: - sage: from sage.combinat.sloane_functions import recur_gen3 sage: it = recur_gen3(1,1,1,1,1,1) sage: [next(it) for i in range(10)] [1, 1, 1, 3, 5, 9, 17, 31, 57, 105] - >>> from sage.all import * >>> from sage.combinat.sloane_functions import recur_gen3 >>> it = recur_gen3(Integer(1),Integer(1),Integer(1),Integer(1),Integer(1),Integer(1)) >>> [next(it) for i in range(Integer(10))] [1, 1, 1, 3, 5, 9, 17, 31, 57, 105]