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
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
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.
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
OEIS
module.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#
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
AUTHORS:
Jaap Spies (2007-02-04)
- class sage.combinat.sloane_functions.A000004#
Bases:
SloaneSequence
The zero sequence.
INPUT:
n
- non negative 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]
AUTHORS:
Jaap Spies (2006-12-10)
- class sage.combinat.sloane_functions.A000005#
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]
AUTHORS:
Jaap Spies (2006-12-10)
William Stein (2007-01-08)
- class sage.combinat.sloane_functions.A000007#
Bases:
SloaneSequence
The characteristic function of 0: \(a(n) = 0^n\).
INPUT:
n
– non negative 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]
AUTHORS:
Jaap Spies (2007-01-12)
- class sage.combinat.sloane_functions.A000008#
Bases:
SloaneSequence
Number of ways of making change for n cents using coins of 1, 2, 5, 10 cents.
INPUT:
n
– non negative 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]
AUTHOR:
Gaski (2009-05-29)
- class sage.combinat.sloane_functions.A000009#
Bases:
SloaneSequence
Number of partitions of \(n\) into odd parts.
INPUT:
n
– non negative 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]
AUTHOR:
Jaap Spies (2007-01-30)
- cf()#
EXAMPLES:
sage: it = sloane.A000009.cf() sage: [next(it) for i in range(14)] [1, 1, 1, 2, 2, 3, 4, 5, 6, 8, 10, 12, 15, 18]
- list(n)#
EXAMPLES:
sage: sloane.A000009.list(14) [1, 1, 1, 2, 2, 3, 4, 5, 6, 8, 10, 12, 15, 18]
- class sage.combinat.sloane_functions.A000010#
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_phi is a standard Sage function implemented in PARI
INPUT:
n
– non negative 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
AUTHORS:
Jaap Spies (2007-01-12)
- class sage.combinat.sloane_functions.A000012#
Bases:
SloaneSequence
The all 1’s sequence.
INPUT:
n
– non negative integer
OUTPUT:
integer
– function value
EXAMPLES:
sage: a = sloane.A000012; a The all 1's 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]
AUTHORS:
Jaap Spies (2007-01-12)
- class sage.combinat.sloane_functions.A000015#
Bases:
SloaneSequence
Smallest prime power \(\geq n\) (where \(1\) is considered a prime power).
INPUT:
n
– non negative 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
AUTHORS:
Jaap Spies (2007-01-18)
- class sage.combinat.sloane_functions.A000016#
Bases:
SloaneSequence
Sloane’s A000016
INPUT:
n
– non negative 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)<PHONE_NUMBER>53048193000 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]
AUTHORS:
Jaap Spies (2007-01-18)
- class sage.combinat.sloane_functions.A000027#
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
Index n is interpreted as _eval(n):
sage: s[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]
- link = 'http://oeis.org/classic/A000027'#
- class sage.combinat.sloane_functions.A000030#
Bases:
SloaneSequence
Initial digit of \(n\).
INPUT:
n
– non negative 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]
AUTHORS:
Jaap Spies (2007-01-18)
- class sage.combinat.sloane_functions.A000032#
Bases:
SloaneSequence
Lucas numbers (beginning at 2): \(L(n) = L(n-1) + L(n-2)\).
INPUT:
n
– non negative 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)<PHONE_NUMBER>38105766356982006981782561278127 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]
AUTHORS:
Jaap Spies (2007-01-18)
- class sage.combinat.sloane_functions.A000035#
Bases:
SloaneSequence
A simple periodic sequence.
INPUT:
n
– non negative 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]
AUTHORS:
Jaap Spies (2007-02-02)
- class sage.combinat.sloane_functions.A000040#
Bases:
SloaneSequence
The prime numbers.
INPUT:
n
– non negative 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
AUTHORS:
Jaap Spies (2007-01-17)
- class sage.combinat.sloane_functions.A000041#
Bases:
SloaneSequence
\(a(n)\) = number of partitions of \(n\) (the partition numbers).
INPUT:
n
– non negative 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)<PHONE_NUMBER>388 sage: a.list(9) [1, 1, 2, 3, 5, 7, 11, 15, 22]
AUTHORS:
Jaap Spies (2007-01-18)
- class sage.combinat.sloane_functions.A000043#
Bases:
SloaneSequence
Primes \(p\) such that \(2^p - 1\) is prime. \(2^p - 1\) is then called a Mersenne prime.
INPUT:
n
– non negative 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]
AUTHORS:
Jaap Spies (2007-01-26)
- class sage.combinat.sloane_functions.A000045#
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
– non negative 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
AUTHORS:
Jaap Spies (2007-01-13)
- fib()#
Returns 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]
- list(n)#
EXAMPLES:
sage: sloane.A000045.list(10) [0, 1, 1, 2, 3, 5, 8, 13, 21, 34]
- class sage.combinat.sloane_functions.A000069#
Bases:
SloaneSequence
Odious numbers: odd number of 1’s in binary expansion.
INPUT:
n
– non negative integer
OUTPUT:
integer
– function value
EXAMPLES:
sage: a = sloane.A000069; a Odious numbers: odd number of 1's in binary expansion. sage: a(0) 1 sage: a(2) 4 sage: a.list(9) [1, 2, 4, 7, 8, 11, 13, 14, 16]
AUTHORS:
Jaap Spies (2007-02-02)
- class sage.combinat.sloane_functions.A000073#
Bases:
SloaneSequence
Tribonacci numbers: a(n) = a(n-1) + a(n-2) + a(n-3). Starting with 0, 0, 1, …
INPUT:
n
– non negative 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]
AUTHORS:
Jaap Spies (2007-01-19)
- list(n)#
EXAMPLES:
sage: sloane.A000073.list(10) [0, 0, 1, 1, 2, 4, 7, 13, 24, 44]
- class sage.combinat.sloane_functions.A000079#
Bases:
SloaneSequence
Powers of 2: \(a(n) = 2^n\).
INPUT:
n
– non negative 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)<PHONE_NUMBER>228229401496703205376 sage: a.list(9) [1, 2, 4, 8, 16, 32, 64, 128, 256]
AUTHORS:
Jaap Spies (2007-01-18)
- class sage.combinat.sloane_functions.A000085#
Bases:
SloaneSequence
Number of self-inverse permutations on \(n\) letters, also known as involutions; number of Young tableaux with \(n\) cells.
INPUT:
n
– non negative 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]
AUTHORS:
Jaap Spies (2007-02-03)
- class sage.combinat.sloane_functions.A000100#
Bases:
SloaneSequence
INPUT:
n
– non negative 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]
AUTHORS:
Jaap Spies (2007-01-26)
- class sage.combinat.sloane_functions.A000108#
Bases:
SloaneSequence
Catalan numbers: \(C_n = \frac{\binom{2n}{n}}{n+1} = \frac{(2n)!}{n!(n+1)!}\).
Also called Segner numbers.
INPUT:
n
– non negative 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)<PHONE_NUMBER>276492108820 sage: a.list(9) [1, 1, 2, 5, 14, 42, 132, 429, 1430]
AUTHORS:
Jaap Spies (2007-01-12)
- class sage.combinat.sloane_functions.A000110#
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
– non negative 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)<PHONE_NUMBER>6764833658790768841387207826363669686825611466616334637559114497892442622672724044217756306953557882560751 sage: a.list(10) [1, 1, 2, 5, 15, 52, 203, 877, 4140, 21147]
AUTHORS:
Nick Alexander
- class sage.combinat.sloane_functions.A000120#
Bases:
SloaneSequence
1’s-counting sequence: number of 1’s in binary expansion of \(n\).
INPUT:
n
– non negative integer
OUTPUT:
integer
– function value
EXAMPLES:
sage: a = sloane.A000120;a 1's-counting sequence: number of 1's 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]
AUTHORS:
Jaap Spies (2007-01-26)
- f(n)#
EXAMPLES:
sage: [sloane.A000120.f(n) for n in range(10)] [0, 1, 1, 2, 1, 2, 2, 3, 1, 2]
- class sage.combinat.sloane_functions.A000124#
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
– non negative 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]
AUTHORS:
Jaap Spies (2007-01-25)
- class sage.combinat.sloane_functions.A000129#
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
– non negative 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]
AUTHORS:
Jaap Spies (2007-01-25)
- class sage.combinat.sloane_functions.A000142#
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
– non negative 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)<PHONE_NUMBER>47897734345611269596115894272000000000 sage: a.list(9) [1, 1, 2, 6, 24, 120, 720, 5040, 40320]
AUTHORS:
Jaap Spies (2007-01-12)
- class sage.combinat.sloane_functions.A000153#
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
– non negative 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)<PHONE_NUMBER>4498196944 sage: a.list(8) [0, 1, 2, 7, 32, 181, 1214, 9403]
AUTHORS:
Jaap Spies (2007-01-13)
- class sage.combinat.sloane_functions.A000165#
Bases:
SloaneSequence
Double factorial numbers: \((2n)!! = 2^n*n!\).
INPUT:
n
– non negative 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)<PHONE_NUMBER>125828464640000 sage: a.list(9) [1, 2, 8, 48, 384, 3840, 46080, 645120, 10321920]
AUTHORS:
Jaap Spies (2007-01-24)
- class sage.combinat.sloane_functions.A000166#
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 \(n\) 0’s not on a line.
INPUT:
n
– non negative 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)<PHONE_NUMBER>92902121 sage: a.list(9) [1, 0, 1, 2, 9, 44, 265, 1854, 14833]
AUTHORS:
Jaap Spies (2007-01-13)
- class sage.combinat.sloane_functions.A000169#
Bases:
SloaneSequence
Number of labeled rooted trees with \(n\) nodes: \(n^{(n-1)}\).
INPUT:
n
– non negative 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)<PHONE_NUMBER> sage: a.list(11) [1, 2, 9, 64, 625, 7776, 117649, 2097152, 43046721,<PHONE_NUMBER>,<PHONE_NUMBER>1]
AUTHORS:
Jaap Spies (2007-01-26)
- class sage.combinat.sloane_functions.A000203#
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
– non negative 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
AUTHORS:
Jaap Spies (2007-01-13)
- class sage.combinat.sloane_functions.A000204#
Bases:
SloaneSequence
Lucas numbers (beginning with 1): \(L(n) = L(n-1) + L(n-2)\) with \(L(1) = 1\), \(L(2) = 3\).
INPUT:
n
– non negative 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)<PHONE_NUMBER>38105766356982006981782561278127 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
AUTHORS:
Jaap Spies (2007-01-18)
- class sage.combinat.sloane_functions.A000213#
Bases:
SloaneSequence
Tribonacci numbers: a(n) = a(n-1) + a(n-2) + a(n-3). Starting with 1, 1, 1, …
INPUT:
n
– non negative 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]
AUTHORS:
Jaap Spies (2007-01-19)
- list(n)#
EXAMPLES:
sage: sloane.A000213.list(10) [1, 1, 1, 3, 5, 9, 17, 31, 57, 105]
- class sage.combinat.sloane_functions.A000217#
Bases:
SloaneSequence
Triangular numbers: \(a(n) = \binom{n+1}{2} = n(n+1)/2\).
INPUT:
n
– non negative 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]
AUTHORS:
Jaap Spies (2007-01-25)
- class sage.combinat.sloane_functions.A000225#
Bases:
SloaneSequence
\(2^n - 1\).
INPUT:
n
– non negative 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]
AUTHORS:
Jaap Spies (2007-01-25)
- class sage.combinat.sloane_functions.A000244#
Bases:
SloaneSequence
Powers of 3: \(a(n) = 3^n\).
INPUT:
n
– non negative 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]
AUTHORS:
Jaap Spies (2007-01-26)
- class sage.combinat.sloane_functions.A000255#
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
– non negative 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)<PHONE_NUMBER>666898717143 sage: a.list(9) [1, 1, 3, 11, 53, 309, 2119, 16687, 148329]
AUTHORS:
Jaap Spies (2007-01-13)
- class sage.combinat.sloane_functions.A000261#
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
– non negative 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)<PHONE_NUMBER>380914335441 sage: a.list(9) [0, 1, 3, 13, 71, 465, 3539, 30637, 296967]
AUTHORS:
Jaap Spies (2007-01-23)
- class sage.combinat.sloane_functions.A000272#
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,<PHONE_NUMBER>]
AUTHORS:
Jaap Spies (2007-01-26)
- class sage.combinat.sloane_functions.A000290#
Bases:
SloaneSequence
The squares: \(a(n) = n^2\).
INPUT:
n
– non negative 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]
AUTHORS:
Jaap Spies (2007-01-25)
- class sage.combinat.sloane_functions.A000292#
Bases:
SloaneSequence
Tetrahedral (or pyramidal) numbers: \(\binom{n+2}{3} = n(n+1)(n+2)/6\).
INPUT:
n
– non negative 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]
AUTHORS:
Jaap Spies (2007-01-26)
- class sage.combinat.sloane_functions.A000302#
Bases:
SloaneSequence
Powers of 4: \(a(n) = 4^n\).
INPUT:
n
– non negative 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]
AUTHORS:
Jaap Spies (2007-01-26)
- class sage.combinat.sloane_functions.A000312#
Bases:
SloaneSequence
Number of labeled mappings from \(n\) points to themselves (endofunctions): \(n^n\).
INPUT:
n
– non negative 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,<PHONE_NUMBER>0]
AUTHORS:
Jaap Spies (2007-01-26)
- class sage.combinat.sloane_functions.A000326#
Bases:
SloaneSequence
Pentagonal numbers: \(n(3n-1)/2\).
INPUT:
n
– non negative 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
AUTHORS:
Jaap Spies (2007-01-26)
- class sage.combinat.sloane_functions.A000330#
Bases:
SloaneSequence
Square pyramidal numbers” \(0^2 + 1^2 \cdots n^2 = n(n+1)(2n+1)/6\).
INPUT:
n
– non negative 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]
AUTHORS:
Jaap Spies (2007-01-26)
- class sage.combinat.sloane_functions.A000396#
Bases:
SloaneSequence
Perfect numbers: equal to sum of proper divisors.
INPUT:
n
– non negative 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)<PHONE_NUMBER>28 sage: a.list(7) [6, 28, 496, 8128, 33550336,<PHONE_NUMBER>,<PHONE_NUMBER>28]
AUTHORS:
Jaap Spies (2007-01-25)
- class sage.combinat.sloane_functions.A000578#
Bases:
SloaneSequence
The cubes: \(a(n) = n^3\).
INPUT:
n
– non negative 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]
AUTHORS:
Jaap Spies (2007-01-26)
- class sage.combinat.sloane_functions.A000583#
Bases:
SloaneSequence
Fourth powers: \(a(n) = n^4\).
INPUT:
n
– non negative 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]
AUTHORS:
Jaap Spies (2007-02-04)
- class sage.combinat.sloane_functions.A000587#
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
– non negative 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)<PHONE_NUMBER>56518507969762382254187048845620355238545130875069912944235105204434466095862371032124545552161 sage: a.list(10) [1, -1, 0, 1, 1, -2, -9, -9, 50, 267]
AUTHORS:
Nick Alexander
- class sage.combinat.sloane_functions.A000668#
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
– non negative 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)<PHONE_NUMBER>60469231731687303715884105727
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,<PHONE_NUMBER>]
AUTHORS:
Jaap Spies (2007-01-25)
- class sage.combinat.sloane_functions.A000670#
Bases:
SloaneSequence
Number of preferential arrangements of \(n\) labeled elements; or number of weak orders on \(n\) labeled elements.
INPUT:
n
– non negative 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]
AUTHORS:
Jaap Spies (2007-02-03)
- class sage.combinat.sloane_functions.A000720#
Bases:
SloaneSequence
\(pi(n)\), the number of primes \(\le n\). Sometimes called \(PrimePi(n)\).
INPUT:
n
– non negative 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]
AUTHORS:
Jaap Spies (2007-01-25)
- class sage.combinat.sloane_functions.A000796#
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
AUTHOR:
Jaap Spies (2007-01-30)
- list(n)#
EXAMPLES:
sage: sloane.A000796.list(10) [3, 1, 4, 1, 5, 9, 2, 6, 5, 3]
- pi()#
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]
- class sage.combinat.sloane_functions.A000961#
Bases:
SloaneSequence
Prime powers
INPUT:
n
– non negative 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]
AUTHORS:
Jaap Spies (2007-01-25)
- list(n)#
EXAMPLES:
sage: sloane.A000961.list(10) [1, 2, 3, 4, 5, 7, 8, 9, 11, 13]
- class sage.combinat.sloane_functions.A000984#
Bases:
SloaneSequence
Central binomial coefficients: \(\binom{2n}{n} = \frac {(2n)!} {(n!)^2}\).
INPUT:
n
– non negative 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]
AUTHORS:
Jaap Spies (2007-01-26)
- class sage.combinat.sloane_functions.A001006#
Bases:
SloaneSequence
Motzkin numbers: number of ways of drawing any number of nonintersecting chords among \(n\) points on a circle.
INPUT:
n
– non negative 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]
AUTHORS:
Jaap Spies (2007-02-02)
- class sage.combinat.sloane_functions.A001045#
Bases:
RecurrenceSequence2
Jacobsthal sequence: \(a(n) = a(n-1) + 2a(n-2)\), \(a(0) = 0\) and \(a(1) = 1\).
INPUT:
n
– non negative 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]
AUTHORS:
Jaap Spies (2007-01-26)
- class sage.combinat.sloane_functions.A001055#
Bases:
SloaneSequence
Number of ways of factoring \(n\) with all factors 1.
INPUT:
n
– non negative 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]
AUTHORS:
Jaap Spies (2007-02-04)
- nwf(n, m)#
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
- class sage.combinat.sloane_functions.A001109#
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
– non negative 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)<PHONE_NUMBER>844857898115857393785728383101709300 sage: a.list(9) [0, 1, 6, 35, 204, 1189, 6930, 40391, 235416]
AUTHORS:
Jaap Spies (2007-01-24)
- class sage.combinat.sloane_functions.A001110#
Bases:
RecurrenceSequence
Numbers that are both triangular and square: \(a(n) = 34a(n-1) - a(n-2) + 2\).
INPUT:
n
– non negative 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)<PHONE_NUMBER>6 sage: a(21)<PHONE_NUMBER>511295358038307980025 sage: a.list(8) [0, 1, 36, 1225, 41616, 1413721, 48024900,<PHONE_NUMBER>]
AUTHORS:
Jaap Spies (2007-01-19)
- g(k)#
EXAMPLES:
sage: sloane.A001110.g(2) 2 sage: sloane.A001110.g(1) 0
- link = 'http://oeis.org/classic/A001110'#
- class sage.combinat.sloane_functions.A001147#
Bases:
SloaneSequence
Double factorial numbers: \((2n-1)!! = 1 \cdot 3 \cdot 5 \cdots (2n-1)\).
INPUT:
n
– non negative 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)<PHONE_NUMBER>72877770815625 sage: a.list(9) [1, 1, 3, 15, 105, 945, 10395, 135135, 2027025]
AUTHORS:
Jaap Spies (2007-01-24)
- class sage.combinat.sloane_functions.A001157#
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
– non negative 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]
AUTHORS:
Jaap Spies (2007-01-13)
- class sage.combinat.sloane_functions.A001189#
Bases:
SloaneSequence
Number of degree-n permutations of order exactly 2.
INPUT:
n
– non negative 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]
AUTHORS:
Jaap Spies (2007-02-03)
- class sage.combinat.sloane_functions.A001221#
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
– non negative 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]
AUTHORS:
Jaap Spies (2007-01-19)
- class sage.combinat.sloane_functions.A001222#
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
– non negative 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]
AUTHORS:
Jaap Spies (2007-01-19)
- class sage.combinat.sloane_functions.A001227#
Bases:
SloaneSequence
Number of odd divisors of \(n\).
INPUT:
n
– non negative 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
AUTHORS:
Jaap Spies (2007-01-14)
- class sage.combinat.sloane_functions.A001333#
Bases:
RecurrenceSequence2
Numerators of continued fraction convergents to \(\sqrt 2\).
See also A000129
INPUT:
n
– non negative 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]
AUTHORS:
Jaap Spies (2007-02-01)
- class sage.combinat.sloane_functions.A001358#
Bases:
SloaneSequence
Products of two primes.
These numbers have been called semiprimes (or semi-primes), biprimes or 2-almost primes.
INPUT:
n
– non negative 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]
AUTHORS:
Jaap Spies (2007-01-25)
- list(n)#
EXAMPLES:
sage: sloane.A001358.list(9) [4, 6, 9, 10, 14, 15, 21, 22, 25]
- class sage.combinat.sloane_functions.A001405#
Bases:
SloaneSequence
Central binomial coefficients: \(\binom{n}{\lfloor \frac {n}{ 2} \rfloor}\).
INPUT:
n
– non negative 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]
AUTHORS:
Jaap Spies (2007-01-26)
- class sage.combinat.sloane_functions.A001477#
Bases:
SloaneSequence
The nonnegative integers.
INPUT:
n
– non negative 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]
AUTHORS:
Jaap Spies (2007-01-26)
- class sage.combinat.sloane_functions.A001694#
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
– non negative 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
AUTHORS:
Jaap Spies (2007-01-14)
- is_powerful(n)#
Return
True
if 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:
True
if \(n\) is a powerful number, elseFalse
EXAMPLES:
sage: a = sloane.A001694 sage: a.is_powerful(2500) True sage: a.is_powerful(20) False
AUTHORS:
Jaap Spies (2006-12-07)
- list(n)#
EXAMPLES:
sage: sloane.A001694.list(9) [1, 4, 8, 9, 16, 25, 27, 32, 36]
- class sage.combinat.sloane_functions.A001836#
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_phi is a standard Sage function.
INPUT:
n
– non negative 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
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)
- list(n)#
EXAMPLES:
sage: sloane.A001836.list(9) [53, 83, 158, 263, 293, 368, 578, 683, 743]
- class sage.combinat.sloane_functions.A001906#
Bases:
RecurrenceSequence2
\(F(2n) =\) bisection of Fibonacci sequence: \(a(n)=3a(n-1)-a(n-2)\).
INPUT:
n
– non negative 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]
AUTHORS:
Jaap Spies (2007-01-19)
- class sage.combinat.sloane_functions.A001909#
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)<PHONE_NUMBER>95287415734 sage: a.list(9) [0, 1, 4, 21, 134, 1001, 8544, 81901, 870274]
AUTHORS:
Jaap Spies (2007-01-13)
- class sage.combinat.sloane_functions.A001910#
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 >= 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)<PHONE_NUMBER>1110663023 sage: a.list(9) [0, 1, 5, 31, 227, 1909, 18089, 190435, 2203319]
AUTHORS:
Jaap Spies (2007-01-13)
- class sage.combinat.sloane_functions.A001969#
Bases:
SloaneSequence
Evil numbers: even number of 1’s in binary expansion.
INPUT:
n
– non negative integer
OUTPUT:
integer
– function value
EXAMPLES:
sage: a = sloane.A001969;a Evil numbers: even number of 1's 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]
AUTHORS:
Jaap Spies (2007-02-02)
- class sage.combinat.sloane_functions.A002110#
Bases:
SloaneSequence
Primorial numbers (first definition): product of first \(n\) primes. Sometimes written \(p\#\).
INPUT:
n
– non negative 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)<PHONE_NUMBER>154212639070 sage: a.list(9) [1, 2, 6, 30, 210, 2310, 30030, 510510, 9699690]
AUTHORS:
Jaap Spies (2007-01-25)
- class sage.combinat.sloane_functions.A002113#
Bases:
SloaneSequence
Palindromes in base 10.
INPUT:
n
– non negative 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]
AUTHORS:
Jaap Spies (2007-02-02)
- list(n)#
EXAMPLES:
sage: sloane.A002113.list(15) [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 22, 33, 44, 55]
- class sage.combinat.sloane_functions.A002275#
Bases:
SloaneSequence
Repunits: \(\frac {(10^n - 1)}{9}\). Often denoted by \(R_n\).
INPUT:
n
– non negative 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)<PHONE_NUMBER>1111111111 sage: a.list(9) [0, 1, 11, 111, 1111, 11111, 111111, 1111111, 11111111]
AUTHORS:
Jaap Spies (2007-01-25)
- class sage.combinat.sloane_functions.A002378#
Bases:
SloaneSequence
Oblong (or pronic, or heteromecic) numbers: \(n(n+1)\).
INPUT:
n
– non negative 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]
AUTHORS:
Jaap Spies (2007-01-26)
- class sage.combinat.sloane_functions.A002620#
Bases:
SloaneSequence
Quarter-squares: floor(n/2)*ceiling(n/2). Equivalently, \(\lfloor n^2/4 \rfloor\).
INPUT:
n
– non negative 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]
AUTHORS:
Jaap Spies (2007-01-26)
- class sage.combinat.sloane_functions.A002808#
Bases:
SloaneSequence
The composite numbers: numbers \(n\) of the form \(xy\) for \(x > 1\) and \(y > 1\).
INPUT:
n
– non negative 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]
AUTHORS:
Jaap Spies (2007-01-26)
- list(n)#
EXAMPLES:
sage: sloane.A002808.list(10) [4, 6, 8, 9, 10, 12, 14, 15, 16, 18]
- class sage.combinat.sloane_functions.A003418#
Bases:
SloaneSequence
Least common multiple (or lcm) of \(\{1, 2, \ldots, n\}\).
INPUT:
n
– non negative 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
AUTHOR:
Jaap Spies (2007-01-31)
- class sage.combinat.sloane_functions.A004086#
Bases:
SloaneSequence
Read n backwards (referred to as \(R(n)\) in many sequences).
INPUT:
n
– non negative 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]
AUTHORS:
Jaap Spies (2007-02-02)
- class sage.combinat.sloane_functions.A004526#
Bases:
SloaneSequence
The nonnegative integers repeated.
INPUT:
n
– non negative 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]
AUTHORS:
Jaap Spies (2007-01-26)
- class sage.combinat.sloane_functions.A005100#
Bases:
SloaneSequence
Deficient numbers: \(\sigma(n) < 2n\).
INPUT:
n
– non negative 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]
AUTHORS:
Jaap Spies (2007-01-26)
- list(n)#
EXAMPLES:
sage: sloane.A005100.list(10) [1, 2, 3, 4, 5, 7, 8, 9, 10, 11]
- class sage.combinat.sloane_functions.A005101#
Bases:
SloaneSequence
Abundant numbers (sum of divisors of \(n\) exceeds \(2n\)).
INPUT:
n
– non negative 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]
AUTHORS:
Jaap Spies (2007-01-26)
- list(n)#
EXAMPLES:
sage: sloane.A005101.list(10) [12, 18, 20, 24, 30, 36, 40, 42, 48, 54]
- class sage.combinat.sloane_functions.A005117#
Bases:
SloaneSequence
Square-free numbers
INPUT:
n
– non negative 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]
AUTHORS:
Jaap Spies (2007-01-25)
- list(n)#
EXAMPLES:
sage: sloane.A005117.list(10) [1, 2, 3, 5, 6, 7, 10, 11, 13, 14]
- class sage.combinat.sloane_functions.A005408#
Bases:
SloaneSequence
The odd numbers a(n) = 2n + 1.
INPUT:
n
– non negative 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]
AUTHORS:
Jaap Spies (2007-01-26)
- class sage.combinat.sloane_functions.A005843#
Bases:
SloaneSequence
The even numbers: \(a(n) = 2n\).
INPUT:
n
– non negative 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]
AUTHORS:
Jaap Spies (2007-02-03)
- class sage.combinat.sloane_functions.A006318#
Bases:
SloaneSequence
Large Schroeder numbers.
INPUT:
n
– non negative 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]
AUTHORS:
Jaap Spies (2007-02-03)
- class sage.combinat.sloane_functions.A006530#
Bases:
SloaneSequence
Largest prime dividing \(n\) (with \(a(1)=1\)).
INPUT:
n
– non negative 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]
AUTHORS:
Jaap Spies (2007-01-25)
- class sage.combinat.sloane_functions.A006882#
Bases:
SloaneSequence
Double factorials \(n!!\): \(a(n)=n \cdot a(n-2)\).
INPUT:
n
– non negative 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)<PHONE_NUMBER> sage: a.list(9) [1, 1, 2, 3, 8, 15, 48, 105, 384]
AUTHORS:
Jaap Spies (2007-01-24)
- df()#
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]
- list(n)#
EXAMPLES:
sage: sloane.A006882.list(10) [1, 1, 2, 3, 8, 15, 48, 105, 384, 945]
- class sage.combinat.sloane_functions.A007318#
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
– non negative 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
AUTHORS:
Jaap Spies (2007-01-31)
- keyword = ['nonn', 'tabl', 'nice', 'easy', 'core', 'triangle']#
- class sage.combinat.sloane_functions.A008275#
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
– non negative 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]
AUTHORS:
Jaap Spies (2007-02-02)
- keyword = ['sign', 'tabl', 'nice', 'core', 'triangle']#
- s(n, k)#
EXAMPLES:
sage: sloane.A008275.s(4,2) 11 sage: sloane.A008275.s(5,2) -50 sage: sloane.A008275.s(5,3) 35
- class sage.combinat.sloane_functions.A008277#
Bases:
SloaneSequence
Triangle of Stirling numbers of 2nd kind, \(S2(n,k)\), \(n \ge 1\), \(1 \le k \le n\).
INPUT:
n
– non negative 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]
AUTHORS:
Jaap Spies (2007-01-31)
- keyword = ['nonn', 'tabl', 'nice', 'core', 'triangle']#
- s2(n, k)#
Returns the Stirling number S2(n,k) of the 2nd kind.
EXAMPLES:
sage: sloane.A008277.s2(4,2) 7
- class sage.combinat.sloane_functions.A008683#
Bases:
SloaneSequence
Möbius function \(\mu(n)\).
INPUT:
n
– non negative 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]
AUTHORS:
Jaap Spies (2007-01-13)
- class sage.combinat.sloane_functions.A010060#
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 0’s and 1’s.
INPUT:
n
– non negative 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]
AUTHORS:
Jaap Spies (2007-02-02)
- class sage.combinat.sloane_functions.A015521#
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
– non negative 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)<PHONE_NUMBER>91703339764941 sage: a.list(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#
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
– non negative 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)<PHONE_NUMBER>474529739091481 sage: a.list(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#
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
– non negative 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]
AUTHORS:
Jaap Spies (2007-01-19)
- class sage.combinat.sloane_functions.A015531#
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
– non negative 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)<PHONE_NUMBER>64733924534327040115992228190104 sage: a.list(9) [0, 1, 4, 21, 104, 521, 2604, 13021, 65104]
AUTHORS:
Jaap Spies (2007-01-19)
- class sage.combinat.sloane_functions.A015551#
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
– non negative 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)<PHONE_NUMBER>530059736761484557155863822531970573036 sage: a.list(9) [0, 1, 6, 41, 276, 1861, 12546, 84581, 570216]
AUTHORS:
Jaap Spies (2007-01-19)
- class sage.combinat.sloane_functions.A018252#
Bases:
SloaneSequence
The nonprime numbers, starting with 1.
INPUT:
n
– non negative 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]
AUTHORS:
Jaap Spies (2007-02-04)
- class sage.combinat.sloane_functions.A020639#
Bases:
SloaneSequence
Least prime dividing \(n\) with \(a(1)=1\).
INPUT:
n
– non negative 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]
AUTHORS:
Jaap Spies (2007-01-25)
- list(n)#
EXAMPLES:
sage: sloane.A020639.list(10) [1, 2, 3, 2, 5, 2, 7, 2, 3, 2]
- class sage.combinat.sloane_functions.A046660(offset=1)#
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]
AUTHORS:
Jaap Spies (2007-01-19)
- class sage.combinat.sloane_functions.A049310#
Bases:
SloaneSequence
Triangle of coefficients of Chebyshev’s \(S(n,x)\): \(U(n, \frac x 2)\) polynomials (exponents in increasing order).
INPUT:
n
– non negative 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']
AUTHORS:
Jaap Spies (2007-01-31)
- keyword = ['sign', 'tabl', 'nice', 'easy', 'core', 'triangle']#
- class sage.combinat.sloane_functions.A051959#
Bases:
RecurrenceSequence
Linear second order recurrence. A051959.
INPUT:
n
– non negative 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)<PHONE_NUMBER>2413650 sage: a.list(12) [1, 10, 36, 104, 273, 686, 1688, 4112, 9969, 24114, 58268, 140728]
AUTHORS:
Jaap Spies (2007-01-19)
- g(k)#
EXAMPLES:
sage: sloane.A051959.g(2) 15 sage: sloane.A051959.g(1) 0
- class sage.combinat.sloane_functions.A055790#
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
– non negative 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)<PHONE_NUMBER>7533070441564 sage: a.list(9) [0, 2, 4, 14, 64, 362, 2428, 18806, 165016]
AUTHORS:
Jaap Spies (2007-01-23)
- class sage.combinat.sloane_functions.A061084#
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
– non negative 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']
AUTHORS:
Jaap Spies (2007-01-18)
- keyword = ['sign', 'easy', 'nice']#
- class sage.combinat.sloane_functions.A064553#
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
– non negative 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]
AUTHORS:
Jaap Spies (2007-02-04)
- class sage.combinat.sloane_functions.A079922(offset=1)#
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]
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
AUTHORS:
Jaap Spies (2007-01-14)
- class sage.combinat.sloane_functions.A079923(offset=1)#
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]
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
AUTHORS:
Jaap Spies (2007-01-17)
- class sage.combinat.sloane_functions.A082411#
Bases:
RecurrenceSequence2
Second-order linear recurrence sequence with \(a(n) = a(n-1) + a(n-2)\).
\(a(0) =<PHONE_NUMBER>18\), \(a(1) =<PHONE_NUMBER>1\). 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
– non negative 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)<PHONE_NUMBER>1 sage: a(2)<PHONE_NUMBER>69 sage: a(3)<PHONE_NUMBER>20 sage: a(20)<PHONE_NUMBER>689173 sage: a.list(4) [407389224418,<PHONE_NUMBER>1,<PHONE_NUMBER>69,<PHONE_NUMBER>20]
AUTHORS:
Jaap Spies (2007-01-23)
- class sage.combinat.sloane_functions.A083103#
Bases:
RecurrenceSequence2
Second-order linear recurrence sequence with \(a(n) = a(n-1) + a(n-2)\).
\(a(0) =<PHONE_NUMBER>928802632268715130455793\), \(a(1) =<PHONE_NUMBER>053915111058165141686995\). 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
– non negative 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)<PHONE_NUMBER>053915111058165141686995 sage: a(2)<PHONE_NUMBER>982717743326880272142788 sage: a(3)<PHONE_NUMBER>036632854385045413829783 sage: a.offset 0 sage: a(8)<PHONE_NUMBER>1206651551714764671352204 sage: a(20)<PHONE_NUMBER>4254059531823985143948191708 sage: a.list(4) [1786772701928802632268715130455793,<PHONE_NUMBER>053915111058165141686995,<PHONE_NUMBER>982717743326880272142788,<PHONE_NUMBER>036632854385045413829783]
AUTHORS:
Jaap Spies (2007-01-23)
- class sage.combinat.sloane_functions.A083104#
Bases:
RecurrenceSequence2
Second-order linear recurrence sequence with \(a(n) = a(n-1) + a(n-2)\).
\(a(0) =<PHONE_NUMBER>98274737472200656430763\), \(a(1) =<PHONE_NUMBER>088401971189590305498785\). 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
– non negative 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)<PHONE_NUMBER>175078679851381267428333 sage: a.offset 0 sage: a(8)<PHONE_NUMBER>0834012982120004949074404 sage: a(20)<PHONE_NUMBER>7621826012468849361236300628
AUTHORS:
Jaap Spies (2007-01-23)
- class sage.combinat.sloane_functions.A083105#
Bases:
RecurrenceSequence2
Second-order linear recurrence sequence with \(a(n) = a(n-1) + a(n-2)\).
\(a(0) =<PHONE_NUMBER>4239857\), \(a(1) =<PHONE_NUMBER>3205655\). 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
– non negative 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)<PHONE_NUMBER>3205655 sage: a(2)<PHONE_NUMBER>47445512 sage: a(3)<PHONE_NUMBER>90651167 sage: a.offset 0 sage: a(8)<PHONE_NUMBER>662436896 sage: a(20)<PHONE_NUMBER>00513098192 sage: a.list(4) [62638280004239857,<PHONE_NUMBER>3205655,<PHONE_NUMBER>47445512,<PHONE_NUMBER>90651167]
AUTHORS:
Jaap Spies (2007-01-23)
- class sage.combinat.sloane_functions.A083216#
Bases:
RecurrenceSequence2
Second-order linear recurrence sequence with \(a(n) = a(n-1) + a(n-2)\).
\(a(0) =<PHONE_NUMBER>5555510\), \(a(1) =<PHONE_NUMBER>567513\). 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
– non negative 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)<PHONE_NUMBER>5555510 sage: a(1)<PHONE_NUMBER>567513 sage: a(8)<PHONE_NUMBER>65139403 sage: a(41)<PHONE_NUMBER>211084205003383 sage: a.list(4) [20615674205555510,<PHONE_NUMBER>567513,<PHONE_NUMBER>7123023,<PHONE_NUMBER>8690536]
AUTHORS:
Jaap Spies (2007-01-19)
- class sage.combinat.sloane_functions.A090010#
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
– non negative 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)<PHONE_NUMBER>29158989003270143 sage: a.list(9) [6, 43, 356, 3333, 34754, 398959, 4996032, 67741129, 988344062]
AUTHORS:
Jaap Spies (2007-01-23)
- class sage.combinat.sloane_functions.A090012#
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
– non negative 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)<PHONE_NUMBER>52297208795129 sage: a.list(9) [3, 9, 39, 213, 1395, 10617, 91911, 890901, 9552387]
AUTHORS:
Jaap Spies (2007-01-23)
- class sage.combinat.sloane_functions.A090013#
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
– non negative 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)<PHONE_NUMBER>171142607670336 sage: a.list(9) [4, 16, 84, 536, 4004, 34176, 327604, 3481096, 40585284]
AUTHORS:
Jaap Spies (2007-01-23)
- class sage.combinat.sloane_functions.A090014#
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
– non negative 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)<PHONE_NUMBER>354446865509725 sage: a.list(9) [5, 25, 155, 1135, 9545, 90445, 952175, 11016595, 138864365]
AUTHORS:
Jaap Spies (2007-01-23)
- class sage.combinat.sloane_functions.A090015#
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
– non negative 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)<PHONE_NUMBER>7692116794936876 sage: a.list(9) [6, 36, 258, 2136, 19998, 208524, 2393754, 29976192, 406446774]
AUTHORS:
Jaap Spies (2007-01-23)
- class sage.combinat.sloane_functions.A090016#
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
– non negative 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)<PHONE_NUMBER>48774341802426289 sage: a.list(9) [7, 49, 399, 3689, 38087, 433713, 5394991, 72737161,<PHONE_NUMBER>]
AUTHORS:
Jaap Spies (2007-01-23)
- class sage.combinat.sloane_functions.A109814#
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
– non negative 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]
AUTHORS:
Jaap Spies (2007-01-13)
- class sage.combinat.sloane_functions.A111774#
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
– non negative 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
AUTHORS:
Jaap Spies (2007-01-13)
- is_number_of_the_third_kind(n)#
Return
True
if 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:
True
if \(n\) is not prime and not a power of 2EXAMPLES:
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
AUTHORS:
Jaap Spies (2006-12-09)
- list(n)#
EXAMPLES:
sage: sloane.A111774.list(12) [6, 9, 10, 12, 14, 15, 18, 20, 21, 22, 24, 25]
- class sage.combinat.sloane_functions.A111775#
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
– non negative 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.
sage: a(1) 0 sage: a(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
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
AUTHORS:
Jaap Spies (2006-12-09)
- class sage.combinat.sloane_functions.A111787#
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
– non negative 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
AUTHORS:
Jaap Spies (2007-01-14)
- class sage.combinat.sloane_functions.ExponentialNumbers(a)#
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
- class sage.combinat.sloane_functions.ExtremesOfPermanentsSequence(offset=1)#
Bases:
SloaneSequence
- gen(a0, a1, d)#
EXAMPLES:
sage: it = sloane.A000153.gen(0,1,2) sage: [next(it) for i in range(5)] [0, 1, 2, 7, 32]
- list(n)#
EXAMPLES:
sage: sloane.A000153.list(8) [0, 1, 2, 7, 32, 181, 1214, 9403]
- class sage.combinat.sloane_functions.ExtremesOfPermanentsSequence2(offset=1)#
Bases:
ExtremesOfPermanentsSequence
- gen(a0, a1, d)#
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]
- class sage.combinat.sloane_functions.RecurrenceSequence(offset=1)#
Bases:
SloaneSequence
- list(n)#
EXAMPLES:
sage: sloane.A001110.list(8) [0, 1, 36, 1225, 41616, 1413721, 48024900,<PHONE_NUMBER>]
- class sage.combinat.sloane_functions.RecurrenceSequence2(offset=1)#
Bases:
SloaneSequence
- list(n)#
EXAMPLES:
sage: sloane.A001906.list(10) [0, 1, 3, 8, 21, 55, 144, 377, 987, 2584]
- class sage.combinat.sloane_functions.Sloane#
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
Ensure none are being incorrectly returned:
sage: [ 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
Ensure that we can access other functions in parent classes:
sage: sloane.__class__ <class 'sage.combinat.sloane_functions.Sloane'>
AUTHORS:
Nick Alexander
- class sage.combinat.sloane_functions.SloaneSequence(offset=1)#
Bases:
SageObject
Base class for a Sloane integer sequence.
- list(n)#
Return
n
terms of the sequence:sequence[offset], sequence[offset+1], ..., sequence[offset+n-1].
EXAMPLES:
sage: sloane.A000012.list(4) [1, 1, 1, 1]
- sage.combinat.sloane_functions.perm_mh(m, h)#
This functions calculates \(f(g,h)\) from Sloane’s sequences A079908-A079928
INPUT:
m
– positive integerh
– non negative 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
AUTHORS:
Jaap Spies (2006)
- sage.combinat.sloane_functions.recur_gen2(a0, a1, a2, a3)#
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]
- sage.combinat.sloane_functions.recur_gen2b(a0, a1, a2, a3, b)#
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]
- sage.combinat.sloane_functions.recur_gen3(a0, a1, a2, a3, a4, a5)#
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]


Lambda Calculus and Types: 2016-2017
|
Lecturer |
|
|
Degrees |
Schedule B2 (CS&P) — Computer Science and Philosophy Schedule B2 — Computer Science Schedule B2 — Mathematics and Computer Science |
|
Term |
Hilary Term 2017 (16 lectures) |
Overview
As a language for describing functions, any literate computer scientist would expect to understand the vocabulary of the lambda calculus. It is folklore that various forms of the lambda calculus are the prototypical functional programming languages, but the pure theory of the lambda calculus is also extremely attractive in its own right. This course introduces the terminology and philosophy of the lambda calculus, and then covers a range of self-contained topics studying the language and some related structures. Topics covered include the equational theory, term rewriting and reduction strategies, combinatory logic, Turing completeness and type systems. As such, the course will also function as a brief introduction to many facets of theoretical computer science, illustrating each (and showing the connections with practical computer science) by its relation to the lambda calculus.There are no prerequisites, but the course will assume familiarity with construting mathematical proofs. Some basic knowledge of computability would be useful for one of the topics (the Models of Computation course is much more than enough), but is certainly not necessary.
Learning outcomes
The course is an introductory overview of the foundations of computer science with particular reference to the lambda-calculus. Students will
- understand the syntax and equational theory of the untyped lambda-calculus, and gain familiarity with manipulation of terms;
- be exposed to a variety of inductive proofs over recursive structures;
- learn techniques for analysing term rewriting systems, with particular reference to beta-reduction;
- see the connections between lambda-calculus and computabilty, and an example of how an undecidability proof can be constructed;
- see the connections and distinctions between lambda-calculus and combinatory logic;
- learn about simple type systems for the lambda-calculus, and how to prove a strong normalization result;
- understand how to deduce types for terms, and prove correctness of a principal type algorithm.
Synopsis
Chapter 0 (1 lecture)
Introductory lecture. Preparation for use of inductive definitions and proofs.
Chapters 1-3 (5 lectures)
Terms, free and bound variables, alpha-conversion, substitution, variable
convention, contexts, the formal theory lambda beta, the eta rule, fixed point combinators, lambda-theories. Reduction.
Compatible closure, reflexive transitive closure, diamond and Church-Rosser properties for general notions of reduction. beta-reduction,
proof of the Church-Rosser property (via parallel reduction), connection between beta-reduction and lambda beta, consistency
of lambda beta. Inconsistency of equating all terms without beta-normal form. Reduction strategies, head and leftmost reduction.
Standard reductions. Proof that leftmost reduction is normalising. Statement, without proof, of Genericity Lemma, and simple
applications.
Chapter 4 (2 lectures)
Church numerals, definability of total recursive functions. Second Recursion
Theorem, Scott-Curry Theorem, undecideability of equality in lambda beta. Briefly, extension to partial functions.
Chapter 5 (2 lectures)
Untyped combinatory algebras. Abstraction algorithm, combinatory completeness,
translations to and from untyped lambda-calculus, mismatches between combinary logic and lambda-calculus, basis. Term algebras.
Chapters 6-8 (6 lectures)
Simple type assignment a la Curry using Hindley's TA lambda system. Contexts
and deductions. Subject Construction Lemma, Subject Reduction Theorem and failure of Subject Expansion. Briefly, a system
with type invariance under equality. Tait's proof of strong normalisation. Consequences: no fixed point combinators, poor
definability power. Pointer to literature on PCF as the obvious extension of simple types to cover all computable functions.
Type substitutions and unification, Robinson's algorithm, the matching problem. Principal Type algorithm and correctness.
Syllabus
Terms, formal theories lambda beta and lambda beta eta, fixed point combinators; reduction, Church-Rosser property of beta-reduction and consistency of lambda beta; reduction strategies, standard reduction sequences, proof that leftmost reduction is normalising; Church numerals, definability of total recursive functions in the lambda-calculus, Second Recusion Theorem and undecidability results; combinatory algebras, combinatory completeness, basis; simple types a la Curry, type deductions, Subject Reduction Theorem, strong normalisation and consequences; type substitutions, unification, matching, correctness of Principal Type AlgorithmReading list
Essential
- Andrew Ker, lecture notes. Available in course materials section. Comprehensive notes on the entire course, including practice questions and class exercises.
Useful Background
- H. P. Barendregt, The Lambda Calculus. North-Holland, revised edition, 1984.
- J. R. Hindley, Basic Simple Type Theory, CUP Cambridge Tracts in Theoretical Computer Science 42, 1997.
- J.-Y. Girard, Y. Lafont, & P. Taylor, Proofs and Types, CUP Cambridge Tracts in Theoretical Computer Science 7, 1989.
- C. Hankin, Lambda Calculi, A Guide for Computer Scientists, OUP Graduate Texts in Computer Science, 1994.
- J. R. Hindley & J. P. Seldin, Introduction to Combinators and Lambda-Calculus, Cambridge University Press, 1986.


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Any studio visiting
10 days with personal trainer
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Any studio visiting
10 days with personal trainer
All day classes
8 days shifting
$150 / person


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