Module: sage.combinat.sloane_functions
Functions that compute some of the sequences in Sloane's tables
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, long, 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.
TESTS:
sage: a = sloane.A000001; sage: a == loads(dumps(a)) True
Author Log:
Module-level Functions
m, h) |
This functions calculates
from Sloane's sequences A079908-A079928
Input:
sage: from sage.combinat.sloane_functions import perm_mh sage: perm_mh(3,3) 36 sage: perm_mh(3,4) 76
Author: Jaap Spies (2006)
a0, a1, a2, a3) |
homogenous 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)
sage: from sage.combinat.sloane_functions import recur_gen2 sage: it = recur_gen2(1,1,1,1) sage: [it.next() for i in range(10)] [1, 1, 2, 3, 5, 8, 13, 21, 34, 55]
a0, a1, a2, a3, b) |
inhomogenous second-order linear recurrence generator with fixed coefficients
and
,
,
.
sage: from sage.combinat.sloane_functions import recur_gen2b sage: it = recur_gen2b(1,1,1,1, lambda n: 0) sage: [it.next() for i in range(10)] [1, 1, 2, 3, 5, 8, 13, 21, 34, 55]
a0, a1, a2, a3, a4, a5) |
homogenous 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)
sage: from sage.combinat.sloane_functions import recur_gen3 sage: it = recur_gen3(1,1,1,1,1,1) sage: [it.next() for i in range(10)] [1, 1, 1, 3, 5, 9, 17, 31, 57, 105]
Class: A000001
self) |
Number of groups of order
.
Note: The database_gap-4.4.9 must be installed for
.
run sage -i database_gap-4.4.9
or higher first.
Input:
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) #optional database_gap 1 sage: a(2) #optional database_gap 1 sage: a(9) #optional database_gap 2 sage: a.list(16) #optional database_gap [1, 1, 1, 2, 1, 2, 1, 5, 2, 2, 1, 5, 1, 2, 1, 14] sage: a(60) # optional 13
Author: Jaap Spies (2007-02-04)
Special Functions: __init__,
_eval,
_repr_
self, n) |
sage: sloane.A000001._eval(4) 2 sage: sloane.A000001._eval(51) #optional requires database_gap
self) |
sage: sloane.A000001._repr_() 'Number of groups of order n.'
Class: A000004
self) |
The zero sequence.
Input:
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]
Author: Jaap Spies (2006-12-10)
Special Functions: __init__,
_eval,
_repr_
self, n) |
sage: sloane.A000004._eval(5) 0
self) |
sage: sloane.A000004._repr_() 'The zero sequence.'
Class: A000005
self) |
The sequence
, which is the number of divisors of
.
This sequence is also denoted
(also called
or
), the number of divisors of n.
Input:
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]
Author Log:
Special Functions: __init__,
_eval,
_repr_
self, n) |
sage: sloane.A000005._eval(5) 2
self) |
sage: sloane.A000005._repr_() 'The integer sequence tau(n), which is the number of divisors of n.'
Class: A000007
self) |
The characteristic function of 0:
.
Input:
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]
Author: Jaap Spies (2007-01-12)
Special Functions: __init__,
_eval,
_repr_
self, n) |
sage: [sloane.A000007._eval(n) for n in range(10)] [1, 0, 0, 0, 0, 0, 0, 0, 0, 0]
self) |
sage: sloane.A000007._repr_() 'The characteristic function of 0: a(n) = 0^n.'
Class: A000009
self) |
Number of partitions of
into odd parts.
Input:
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)
Functions: cf,
list
self) |
sage: it = sloane.A000009.cf() sage: [it.next() for i in range(14)] [1, 1, 1, 2, 2, 3, 4, 5, 6, 8, 10, 12, 15, 18]
self, n) |
sage: sloane.A000009.list(14) [1, 1, 1, 2, 2, 3, 4, 5, 6, 8, 10, 12, 15, 18]
Special Functions: __init__,
_eval,
_precompute,
_repr_
self, n) |
sage: [sloane.A000009._eval(i) for i in range(14)] [1, 1, 1, 2, 2, 3, 4, 5, 6, 8, 10, 12, 15, 18]
self, [how_many=50]) |
sage: initial = len(sloane.A000009._b) sage: sloane.A000009._precompute(10) sage: len(sloane.A000009._b) - initial == 10 True
self) |
sage: sloane.A000009._repr_() 'Number of partitions of n into odd parts.'
Class: A000010
self) |
The integer sequence A000010 is Euler's totient function.
Number of positive integers
that are relative prime to
.
Number of totatives of
.
Euler totient function
: count numbers <
and prime to
.
euler_phi is a standard SAGE function implemented in PARI
Input:
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, long, or Integer
Author: Jaap Spies (2007-01-12)
Special Functions: __init__,
_eval,
_repr_
self, n) |
sage: [sloane.A000010._eval(n) for n in range(1,11)] [1, 1, 2, 2, 4, 2, 6, 4, 6, 4]
self) |
sage: sloane.A000010._repr_() "Euler's totient function"
Class: A000012
self) |
The all 1's sequence.
Input:
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]
Author: Jaap Spies (2007-01-12)
Special Functions: __init__,
_eval,
_repr_
self, n) |
sage: [sloane.A000012._eval(n) for n in range(10)] [1, 1, 1, 1, 1, 1, 1, 1, 1, 1]
self) |
sage: sloane.A000012._repr_() "The all 1's sequence."
Class: A000015
self) |
Smallest prime power
.
Input:
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
Author: Jaap Spies (2007-01-18)
Special Functions: __init__,
_eval,
_repr_
self, n) |
sage: [sloane.A000015._eval(n) for n in range(1,11)] [1, 2, 3, 4, 5, 7, 7, 8, 9, 11]
self) |
sage: sloane.A000015._repr_() 'Smallest prime power >= n.'
Class: A000016
self) |
Sloane's A000016
Input:
sage: a = sloane.A000016; a Sloane's A000016. sage: a(1) 1 sage: a(0) 1 sage: a(8) 16 sage: a(75) 251859545753048193000 sage: a(-4) Traceback (most recent call last): ... ValueError: input n (=-4) must be an integer >= 0 sage: a.list(12) [1, 1, 1, 2, 2, 4, 6, 10, 16, 30, 52, 94]
Author: Jaap Spies (2007-01-18)
Special Functions: __init__,
_eval,
_repr_
self, n) |
sage: [sloane.A000016._eval(n) for n in range(10)] [1, 1, 1, 2, 2, 4, 6, 10, 16, 30]
self) |
sage: sloane.A000016._repr_() "Sloane's A000016."
Class: A000027
self) |
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.
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]
Special Functions: __init__,
_eval,
_repr_
self, n) |
sage: sloane.A000027._eval(5) 5
self) |
sage: sloane.A000027._repr_() 'The natural numbers.'
Class: A000030
self) |
Initial digit of
.
Input:
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]
Author: Jaap Spies (2007-01-18)
Special Functions: __init__,
_eval,
_repr_
self, n) |
sage: [sloane.A000030._eval(n) for n in range(10)] [0, 1, 2, 3, 4, 5, 6, 7, 8, 9]
self) |
sage: sloane.A000030._repr_() 'Initial digit of n'
Class: A000032
self) |
Lucas numbers (beginning at 2):
.
Input:
sage: a = sloane.A000032; a Lucas numbers (beginning at 2): L(n) = L(n-1) + L(n-2). sage: a(0) 2 sage: a(1) 1 sage: a(8) 47 sage: a(200) 627376215338105766356982006981782561278127 sage: a(-4) Traceback (most recent call last): ... ValueError: input n (=-4) must be an integer >= 0 sage: a.list(12) [2, 1, 3, 4, 7, 11, 18, 29, 47, 76, 123, 199]
Author: Jaap Spies (2007-01-18)
Special Functions: __init__,
_eval,
_repr_
self, n) |
sage: [sloane.A000032._eval(n) for n in range(10)] [2, 1, 3, 4, 7, 11, 18, 29, 47, 76]
self) |
sage: sloane.A000032._repr_() 'Lucas numbers (beginning at 2): L(n) = L(n-1) + L(n-2).'
Class: A000035
self) |
A simple periodic sequence.
Input:
sage: a = sloane.A000035;a A simple periodic sequence. sage: a(0.0) Traceback (most recent call last): ... TypeError: input must be an int, long, 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]
Author: Jaap Spies (2007-02-02)
Special Functions: __init__,
_eval,
_repr_
self, n) |
sage: [sloane.A000035._eval(n) for n in range(10)] [0, 1, 0, 1, 0, 1, 0, 1, 0, 1]
self) |
sage: sloane.A000035._repr_() 'A simple periodic sequence.'
Class: A000040
self) |
The prime numbers.
Input:
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
Author: Jaap Spies (2007-01-17)
Special Functions: __init__,
_eval,
_repr_
self, n) |
sage: [sloane.A000040._eval(n) for n in range(1,11)] [2, 3, 5, 7, 11, 13, 17, 19, 23, 29]
self) |
sage: sloane.A000040._repr_() 'The prime numbers.'
Class: A000041
self) |
= number of partitions of
(the partition numbers).
Input:
sage: a = sloane.A000041;a a(n) = number of partitions of n (the partition numbers). sage: a(0) 1 sage: a(2) 2 sage: a(8) 22 sage: a(200) 3972999029388 sage: a.list(9) [1, 1, 2, 3, 5, 7, 11, 15, 22]
Author: Jaap Spies (2007-01-18)
Special Functions: __init__,
_eval,
_repr_
self, n) |
sage: [sloane.A000041._eval(n) for n in range(1,11)] [1, 2, 3, 5, 7, 11, 15, 22, 30, 42]
self) |
sage: sloane.A000041._repr_() 'a(n) = number of partitions of n (the partition numbers).'
Class: A000043
self) |
Primes
such that
is prime.
is then called a Mersenne prime.
Input:
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]
Author: Jaap Spies (2007-01-26)
Special Functions: __init__,
_eval,
_repr_
self, n) |
sage: [sloane.A000043._eval(n) for n in range(1,11)] [2, 3, 5, 7, 13, 17, 19, 31, 61, 89]
self) |
sage: sloane.A000043._repr_() 'Primes p such that 2^p - 1 is prime. 2^p - 1 is then called a Mersenne prime.'
Class: A000045
self) |
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:
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, long, or Integer
Author: Jaap Spies (2007-01-13)
Functions: fib,
list
self) |
Returns a generator over all Fibanacci numbers, starting with 0.
sage: it = sloane.A000045.fib() sage: [it.next() for i in range(10)] [0, 1, 1, 2, 3, 5, 8, 13, 21, 34]
self, n) |
sage: sloane.A000045.list(10) [0, 1, 1, 2, 3, 5, 8, 13, 21, 34]
Special Functions: __init__,
_eval,
_precompute,
_repr_
self, n) |
sage: [sloane.A000045._eval(n) for n in range(1,11)] [1, 1, 2, 3, 5, 8, 13, 21, 34, 55]
self, [how_many=500]) |
sage: initial = len(sloane.A000045._b) sage: sloane.A000045._precompute(10) sage: len(sloane.A000045._b) - initial > 0 True
self) |
sage: sloane.A000045._repr_() 'Fibonacci numbers with index n >= 0'
Class: A000069
self) |
Odious numbers: odd number of 1's in binary expansion.
Input:
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]
Author: Jaap Spies (2007-02-02)
Special Functions: __init__,
_eval,
_repr_
self, n) |
sage: [sloane.A000069._eval(n) for n in range(10)] [1, 2, 4, 7, 8, 11, 13, 14, 16, 19]
self) |
sage: sloane.A000069._repr_() "Odious numbers: odd number of 1's in binary expansion."
Class: A000073
self) |
Tribonacci numbers: a(n) = a(n-1) + a(n-2) + a(n-3). Starting with 0, 0, 1, ...
Input:
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]
Author: Jaap Spies (2007-01-19)
Functions: list
self, n) |
sage: sloane.A000073.list(10) [0, 0, 1, 1, 2, 4, 7, 13, 24, 44]
Special Functions: __init__,
_eval,
_precompute,
_repr_
self, n) |
sage: [sloane.A000073._eval(n) for n in range(10)] [0, 0, 1, 1, 2, 4, 7, 13, 24, 44]
self, [how_many=20]) |
sage: initial = len(sloane.A000073._b) sage: sloane.A000073._precompute(10) sage: len(sloane.A000073._b) - initial == 10 True
self) |
sage: sloane.A000073._repr_() 'Tribonacci numbers: a(n) = a(n-1) + a(n-2) + a(n-3).'
Class: A000079
self) |
Powers of 2:
.
Input:
sage: a = sloane.A000079;a Powers of 2: a(n) = 2^n. sage: a(0) 1 sage: a(2) 4 sage: a(8) 256 sage: a(100) 1267650600228229401496703205376 sage: a.list(9) [1, 2, 4, 8, 16, 32, 64, 128, 256]
Author: Jaap Spies (2007-01-18)
Special Functions: __init__,
_eval,
_repr_
self, n) |
sage: [sloane.A000079._eval(n) for n in range(10)] [1, 2, 4, 8, 16, 32, 64, 128, 256, 512]
self) |
sage: sloane.A000079._repr_() 'Powers of 2: a(n) = 2^n.'
Class: A000085
self) |
Number of self-inverse permutations on
letters, also known as involutions; number of Young tableaux with
cells.
Input:
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]
Author: Jaap Spies (2007-02-03)
Special Functions: __init__,
_eval,
_repr_
self, n) |
sage: [sloane.A000085._eval(n) for n in range(10)] [1, 1, 2, 4, 10, 26, 76, 232, 764, 2620]
self) |
sage: sloane.A000085._repr_() 'Number of self-inverse permutations on n letters.'
Class: A000100
self) |
Input:
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]
Author: Jaap Spies (2007-01-26)
Special Functions: __init__,
_eval,
_repr_
self, n) |
sage: [sloane.A000100._eval(n) for n in range(10)] [0, 0, 0, 1, 2, 5, 11, 23, 47, 94]
self) |
sage: sloane.A000100._repr_() 'Number of compositions of n in which the maximum part size is 3.'
Class: A000108
self) |
Catalan numbers:
. Also called Segner numbers.
Input:
sage: a = sloane.A000108;a Catalan numbers: C(n) = binomial(2n,n)/(n+1) = (2n)!/(n!(n+1)!). Also called Segner numbers. sage: a(0) 1 sage: a.offset 0 sage: a(8) 1430 sage: a(40) 2622127042276492108820 sage: a.list(9) [1, 1, 2, 5, 14, 42, 132, 429, 1430]
Author: Jaap Spies (2007-01-12)
Special Functions: __init__,
_eval,
_repr_
self, n) |
sage: [sloane.A000108._eval(n) for n in range(10)] [1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862]
self) |
sage: sloane.A000108._repr_() 'Catalan numbers: C(n) = binomial(2n,n)/(n+1) = (2n)!/(n!(n+1)!). Also called Segner numbers.'
Class: A000110
self) |
The sequence of Bell numbers.
The Bell number
counts the number of ways to put
distinguishable things into indistinguishable boxes such that no
box is empty.
Let
denote the Stirling number of the second kind. Then
Input:
sage: a = sloane.A000110; a Sequence of Bell numbers sage: a.offset 0 sage: a(0) 1 sage: a(100) 475853912767648336587907688413872078263636696868256114666163346375591144978 92442622672724044217756306953557882560751 sage: a.list(10) [1, 1, 2, 5, 15, 52, 203, 877, 4140, 21147]
Author: Nick Alexander
Special Functions: __init__,
_repr_
self) |
sage: sloane.A000110._repr_() 'Sequence of Bell numbers'
Class: A000120
self) |
1's-counting sequence: number of 1's in binary expansion of
.
Input:
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]
Author: Jaap Spies (2007-01-26)
Functions: f
self, n) |
sage: [sloane.A000120.f(n) for n in range(10)] [0, 1, 1, 2, 1, 2, 2, 3, 1, 2]
Special Functions: __init__,
_eval,
_repr_
self, n) |
sage: [sloane.A000120._eval(n) for n in range(10)] [0, 1, 1, 2, 1, 2, 2, 3, 1, 2]
self) |
sage: sloane.A000120._repr_() "1's-counting sequence: number of 1's in binary expansion of n."
Class: A000124
self) |
Central polygonal numbers (the Lazy Caterer's sequence):
.
Or, maximal number of pieces formed when slicing a pancake with
cuts.
Input:
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]
Author: Jaap Spies (2007-01-25)
Special Functions: __init__,
_eval,
_repr_
self, n) |
sage: [sloane.A000124._eval(n) for n in range(10)] [1, 2, 4, 7, 11, 16, 22, 29, 37, 46]
self) |
sage: sloane.A000124._repr_() "Central polygonal numbers (the Lazy Caterer's sequence): n(n+1)/2 + 1."
Class: A000129
self) |
Pell numbers:
,
; for
,
.
Denominators of continued fraction convergents to
.
See also A001333
Input:
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]
Author: Jaap Spies (2007-01-25)
Special Functions: __init__,
_repr_
self) |
sage: sloane.A000129._repr_() 'Pell numbers: a(0) = 0, a(1) = 1; for n > 1, a(n) = 2*a(n-1) + a(n-2).'
Class: A000142
self) |
Factorial numbers:
Order of symmetric group
, number of permutations of
letters.
Input:
sage: a = sloane.A000142;a Factorial numbers: n! = 1*2*3*4*...*n (order of symmetric group S_n, number of permutations of n letters). sage: a(0) 1 sage: a(8) 40320 sage: a(40) 815915283247897734345611269596115894272000000000 sage: a.list(9) [1, 1, 2, 6, 24, 120, 720, 5040, 40320]
Author: Jaap Spies (2007-01-12)
Special Functions: __init__,
_eval,
_repr_
self, n) |
sage: [sloane.A000142._eval(n) for n in range(10)] [1, 1, 2, 6, 24, 120, 720, 5040, 40320, 362880]
self) |
sage: sloane.A000142._repr_() 'Factorial numbers: n! = 1*2*3*4*...*n (order of symmetric group S_n, number of permutations of n letters).'
Class: A000153
self) |
, with
,
.
With offset 1, permanent of (0,1)-matrix of size
with
and
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:
sage: a = sloane.A000153; a a(n) = n*a(n-1) + (n-2)*a(n-2), with a(0) = 0, a(1) = 1. sage: a(0) 0 sage: a(1) 1 sage: a(8) 82508 sage: a(20) 10315043624498196944 sage: a.list(8) [0, 1, 2, 7, 32, 181, 1214, 9403]
Author: Jaap Spies (2007-01-13)
Special Functions: __init__,
_repr_
self) |
sage: sloane.A000153._repr_() 'a(n) = n*a(n-1) + (n-2)*a(n-2), with a(0) = 0, a(1) = 1.'
Class: A000165
self) |
Double factorial numbers:
.
Input:
sage: a = sloane.A000165;a Double factorial numbers: (2n)!! = 2^n*n!. sage: a(0) 1 sage: a.offset 0 sage: a(8) 10321920 sage: a(20) 2551082656125828464640000 sage: a.list(9) [1, 2, 8, 48, 384, 3840, 46080, 645120, 10321920]
Author: Jaap Spies (2007-01-24)
Special Functions: __init__,
_eval,
_repr_
self, n) |
sage: [sloane.A000165._eval(n) for n in range(10)] [1, 2, 8, 48, 384, 3840, 46080, 645120, 10321920, 185794560]
self) |
sage: sloane.A000165._repr_() 'Double factorial numbers: (2n)!! = 2^n*n!.'
Class: A000166
self) |
Subfactorial or rencontres numbers, or derangements: number of permutations of
elements with no fixed points.
With offset 1 also the permanent of a (0,1)-matrix of order
with
0's not on a line.
Input:
sage: a = sloane.A000166;a Subfactorial or rencontres numbers, or derangements: number of permutations of $n$ elements with no fixed points. sage: a(0) 1 sage: a(1) 0 sage: a(2) 1 sage: a.offset 0 sage: a(8) 14833 sage: a(20) 895014631192902121 sage: a.list(9) [1, 0, 1, 2, 9, 44, 265, 1854, 14833]
Author: Jaap Spies (2007-01-13)
Special Functions: __init__,
_eval,
_repr_
self, n) |
sage: [sloane.A000166._eval(n) for n in range(9)] [1, 0, 1, 2, 9, 44, 265, 1854, 14833]
self) |
sage: sloane.A000166._repr_() 'Subfactorial or rencontres numbers, or derangements: number of permutations of $n$ elements with no fixed points.'
Class: A000169
self) |
Number of labeled rooted trees with
nodes:
.
Input:
sage: a = sloane.A000169;a Number of labeled rooted trees with n nodes: n^(n-1). sage: a(0) Traceback (most recent call last): ... ValueError: input n (=0) must be a positive integer sage: a(1) 1 sage: a(2) 2 sage: a(10) 1000000000 sage: a.list(11) [1, 2, 9, 64, 625, 7776, 117649, 2097152, 43046721, 1000000000, 25937424601]
Author: Jaap Spies (2007-01-26)
Special Functions: __init__,
_eval,
_repr_
self, n) |
sage: [sloane.A000169._eval(n) for n in range(1,11)] [1, 2, 9, 64, 625, 7776, 117649, 2097152, 43046721, 1000000000]
self) |
sage: sloane.A000169._repr_() 'Number of labeled rooted trees with n nodes: n^(n-1).'
Class: A000203
self) |
The sequence
, where
is the sum of the
divisors of
. Also called
.
The function sigma(n, k)
implements
in SAGE.
Input:
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, long, or Integer
Author: Jaap Spies (2007-01-13)
Special Functions: __init__,
_eval,
_repr_
self, n) |
sage: [sloane.A000203._eval(n) for n in range(1,11)] [1, 3, 4, 7, 6, 12, 8, 15, 13, 18]
self) |
sage: sloane.A000203._repr_() 'sigma(n) = sum of divisors of n. Also called sigma_1(n).'
Class: A000204
self) |
Lucas numbers (beginning with 1):
with
,
.
Input:
sage: a = sloane.A000204; a Lucas numbers (beginning at 1): L(n) = L(n-1) + L(n-2), L(2) = 3. sage: a(1) 1 sage: a(8) 47 sage: a(200) 627376215338105766356982006981782561278127 sage: a(-4) Traceback (most recent call last): ... ValueError: input n (=-4) must be a positive integer sage: a.list(12) [1, 3, 4, 7, 11, 18, 29, 47, 76, 123, 199, 322] sage: a(0) Traceback (most recent call last): ... ValueError: input n (=0) must be a positive integer
Author: Jaap Spies (2007-01-18)
Special Functions: __init__,
_eval,
_repr_
self, n) |
sage: [sloane.A000204._eval(n) for n in range(1,11)] [1, 3, 4, 7, 11, 18, 29, 47, 76, 123]
self) |
sage: sloane.A000204._repr_() 'Lucas numbers (beginning at 1): L(n) = L(n-1) + L(n-2), L(2) = 3.'
Class: A000213
self) |
Tribonacci numbers: a(n) = a(n-1) + a(n-2) + a(n-3). Starting with 1, 1, 1, ...
Input:
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]
Author: Jaap Spies (2007-01-19)
Functions: list
self, n) |
sage: sloane.A000213.list(10) [1, 1, 1, 3, 5, 9, 17, 31, 57, 105]
Special Functions: __init__,
_eval,
_precompute,
_repr_
self, n) |
sage: [sloane.A000213._eval(n) for n in range(10)] [1, 1, 1, 3, 5, 9, 17, 31, 57, 105]
self, [how_many=20]) |
sage: initial = len(sloane.A000213._b) sage: sloane.A000213._precompute(10) sage: len(sloane.A000213._b) - initial == 10 True
self) |
sage: sloane.A000213._repr_() 'Tribonacci numbers: a(n) = a(n-1) + a(n-2) + a(n-3).'
Class: A000217
self) |
Triangular numbers:
.
Input:
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]
Author: Jaap Spies (2007-01-25)
Special Functions: __init__,
_eval,
_repr_
self, n) |
sage: [sloane.A000217._eval(n) for n in range(10)] [0, 1, 3, 6, 10, 15, 21, 28, 36, 45]
self) |
sage: sloane.A000217._repr_() 'Triangular numbers: a(n) = C(n+1,2) = n(n+1)/2 = 0+1+2+...+n.'
Class: A000225
self) |
.
Input:
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]
Author: Jaap Spies (2007-01-25)
Special Functions: __init__,
_eval,
_repr_
self, n) |
sage: [sloane.A000225._eval(n) for n in range(10)] [0, 1, 3, 7, 15, 31, 63, 127, 255, 511]
self) |
sage: sloane.A000225._repr_() '2^n - 1.'
Class: A000244
self) |
Powers of 3:
.
Input:
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]
Author: Jaap Spies (2007-01-26)
Special Functions: __init__,
_eval,
_repr_
self, n) |
sage: [sloane.A000244._eval(n) for n in range(10)] [1, 3, 9, 27, 81, 243, 729, 2187, 6561, 19683]
self) |
sage: sloane.A000244._repr_() 'Powers of 3: a(n) = 3^n.'
Class: A000255
self) |
, with
,
.
With offset 1, permanent of (0,1)-matrix of size
with
and
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:
sage: a = sloane.A000255;a a(n) = n*a(n-1) + (n-1)*a(n-2), a(0) = 1, a(1) = 1. sage: a(0) 1 sage: a(1) 1 sage: a.offset 0 sage: a(8) 148329 sage: a(22) 9923922230666898717143 sage: a.list(9) [1, 1, 3, 11, 53, 309, 2119, 16687, 148329]
Author: Jaap Spies (2007-01-13)
Special Functions: __init__,
_repr_
self) |
sage: sloane.A000255._repr_() 'a(n) = n*a(n-1) + (n-1)*a(n-2), a(0) = 1, a(1) = 1.'
Class: A000261
self) |
, with
,
.
With offset 1, permanent of (0,1)-matrix of size
with
and
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:
sage: a = sloane.A000261;a a(n) = n*a(n-1) + (n-3)*a(n-2), a(1) = 0, a(2) = 1. sage: a(0) Traceback (most recent call last): ... ValueError: input n (=0) must be a positive integer sage: a(1) 0 sage: a.offset 1 sage: a(8) 30637 sage: a(22) 1801366114380914335441 sage: a.list(9) [0, 1, 3, 13, 71, 465, 3539, 30637, 296967]
Author: Jaap Spies (2007-01-23)
Special Functions: __init__,
_repr_
self) |
sage: sloane.A000261._repr_() 'a(n) = n*a(n-1) + (n-3)*a(n-2), a(1) = 0, a(2) = 1.'
Class: A000272
self) |
Number of labeled rooted trees on
nodes:
.
Input:
sage: a = sloane.A000272;a Number of labeled rooted trees with n nodes: n^(n-2). 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(10) 100000000 sage: a.list(11) [1, 1, 3, 16, 125, 1296, 16807, 262144, 4782969, 100000000, 2357947691]
Author: Jaap Spies (2007-01-26)
Special Functions: __init__,
_eval,
_repr_
self, n) |
sage: [sloane.A000272._eval(n) for n in range(1,11)] [1, 1, 3, 16, 125, 1296, 16807, 262144, 4782969, 100000000]
self) |
sage: sloane.A000272._repr_() 'Number of labeled rooted trees with n nodes: n^(n-2).'
Class: A000290
self) |
The squares:
.
Input:
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]
Author: Jaap Spies (2007-01-25)
Special Functions: __init__,
_eval,
_repr_
self, n) |
sage: [sloane.A000290._eval(n) for n in range(10)] [0, 1, 4, 9, 16, 25, 36, 49, 64, 81]
self) |
sage: sloane.A000290._repr_() 'The squares: a(n) = n^2.'
Class: A000292
self) |
Tetrahedral (or pyramidal) numbers:
.
Input:
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]
Author: Jaap Spies (2007-01-26)
Special Functions: __init__,
_eval,
_repr_
self, n) |
sage: [sloane.A000292._eval(n) for n in range(10)] [0, 1, 4, 10, 20, 35, 56, 84, 120, 165]
self) |
sage: sloane.A000292._repr_() 'Tetrahedral (or pyramidal) numbers: C(n+2,3) = n(n+1)(n+2)/6.'
Class: A000302
self) |
Powers of 4:
.
Input:
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]
Author: Jaap Spies (2007-01-26)
Special Functions: __init__,
_eval,
_repr_
self, n) |
sage: [sloane.A000302._eval(n) for n in range(10)] [1, 4, 16, 64, 256, 1024, 4096, 16384, 65536, 262144]
self) |
sage: sloane.A000302._repr_() 'Powers of 4: a(n) = 4^n.'
Class: A000312
self) |
Number of labeled mappings from
points to themselves (endofunctions):
.
Input:
sage: a = sloane.A000312;a Number of labeled mappings from n points to themselves (endofunctions): n^n. sage: a(-1) Traceback (most recent call last): ... ValueError: input n (=-1) must be an integer >= 0 sage: a(0) 1 sage: a(1) 1 sage: a(9) 387420489 sage: a.list(11) [1, 1, 4, 27, 256, 3125, 46656, 823543, 16777216, 387420489, 10000000000]
Author: Jaap Spies (2007-01-26)
Special Functions: __init__,
_eval,
_repr_
self, n) |
sage: [sloane.A000312._eval(n) for n in range(10)] [1, 1, 4, 27, 256, 3125, 46656, 823543, 16777216, 387420489]
self) |
sage: sloane.A000312._repr_() 'Number of labeled mappings from n points to themselves (endofunctions): n^n.'
Class: A000326
self) |
Pentagonal numbers:
.
Input:
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, long, or Integer
Author: Jaap Spies (2007-01-26)
Special Functions: __init__,
_eval,
_repr_
self, n) |
sage: [sloane.A000326._eval(n) for n in range(10)] [0, 1, 5, 12, 22, 35, 51, 70, 92, 117]
self) |
sage: sloane.A000326._repr_() 'Pentagonal numbers: n(3n-1)/2.'
Class: A000330
self) |
Square pyramidal numbers"
.
Input:
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]
Author: Jaap Spies (2007-01-26)
Special Functions: __init__,
_eval,
_repr_
self, n) |
sage: [sloane.A000330._eval(n) for n in range(10)] [0, 1, 5, 14, 30, 55, 91, 140, 204, 285]
self) |
sage: sloane.A000330._repr_() 'Square pyramidal numbers: 0^2+1^2+2^2+...+n^2 = n(n+1)(2n+1)/6.'
Class: A000396
self) |
Perfect numbers: equal to sum of proper divisors.
Input:
sage: a = sloane.A000396;a Perfect numbers: equal to sum of proper divisors. sage: a(0) Traceback (most recent call last): ... ValueError: input n (=0) must be a positive integer sage: a(1) 6 sage: a(2) 28 sage: a(7) 137438691328 sage: a.list(7) [6, 28, 496, 8128, 33550336, 8589869056, 137438691328]
Author: Jaap Spies (2007-01-25)
Special Functions: __init__,
_eval,
_repr_
self, n) |
sage: [sloane.A000396._eval(n) for n in range(1,6)] [6, 28, 496, 8128, 33550336]
self) |
sage: sloane.A000396._repr_() 'Perfect numbers: equal to sum of proper divisors.'
Class: A000578
self) |
The cubes:
.
Input:
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]
Author: Jaap Spies (2007-01-26)
Special Functions: __init__,
_eval,
_repr_
self, n) |
sage: [sloane.A000578._eval(n) for n in range(10)] [0, 1, 8, 27, 64, 125, 216, 343, 512, 729]
self) |
sage: sloane.A000578._repr_() 'The cubes: n^3'
Class: A000583
self) |
Fourth powers:
.
Input:
sage: a = sloane.A000583;a Fourth powers: n^4. sage: a(0.0) Traceback (most recent call last): ... TypeError: input must be an int, long, 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]
Author: Jaap Spies (2007-02-04)
Special Functions: __init__,
_eval,
_repr_
self, n) |
sage: [sloane.A000583._eval(n) for n in range(10)] [0, 1, 16, 81, 256, 625, 1296, 2401, 4096, 6561]
self) |
sage: sloane.A000583._repr_() 'Fourth powers: n^4.'
Class: A000587
self) |
The sequence of Uppuluri-Carpenter numbers.
The Uppuluri-Carpenter number
counts the imbalance in the
number of ways to put
distinguishable things into an even
number of indistinguishable boxes versus into an odd number of
indistinguishable boxes, such that no box is empty.
Let
denote the Stirling number of the second kind. Then
Input:
sage: a = sloane.A000587; a Sequence of Uppuluri-Carpenter numbers sage: a.offset 0 sage: a(0) 1 sage: a(100) 397577026456518507969762382254187048845620355238545130875069912944235105204 434466095862371032124545552161 sage: a.list(10) [1, -1, 0, 1, 1, -2, -9, -9, 50, 267]
Author: Nick Alexander
Special Functions: __init__,
_repr_
self) |
sage: sloane.A000587._repr_() 'Sequence of Uppuluri-Carpenter numbers'
Class: A000668
self) |
Mersenne primes (of form
where
is a prime).
(See A000043 for the values of
.)
Warning: a(39) has 4,053,946 digits!
Input:
sage: a = sloane.A000668;a Mersenne primes (of form 2^p - 1 where p is a prime). (See A000043 for the values of p.) sage: a(1) 3 sage: a(2) 7 sage: a(12) 170141183460469231731687303715884105727
Warning: a(39) has 4,053,946 digits!
sage: a(40) Traceback (most recent call last): ... IndexError: list index out of range sage: a.list(8) [3, 7, 31, 127, 8191, 131071, 524287, 2147483647]
Author: Jaap Spies (2007-01-25)
Special Functions: __init__,
_eval,
_repr_
self, n) |
sage: [sloane.A000668._eval(n) for n in range(1,11)] [3, 7, 31, 127, 8191, 131071, 524287, 2147483647, 2305843009213693951, 618970019642690137449562111]
self) |
sage: sloane.A000668._repr_() 'Mersenne primes (of form 2^p - 1 where p is a prime). (See A000043 for the values of p.)'
Class: A000670
self) |
Number of preferential arrangements of
labeled elements; or number of weak orders on
labeled elements.
Input:
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]
Author: Jaap Spies (2007-02-03)
Special Functions: __init__,
_eval,
_repr_
self, n) |
sage: [sloane.A000670._eval(n) for n in range(1,10)] [1, 3, 13, 75, 541, 4683, 47293, 545835, 7087261]
self) |
sage: sloane.A000670._repr_() 'Number of preferential arrangements of n labeled elements.'
Class: A000720
self) |
, the number of primes
. Sometimes called
.
Input:
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]
Author: Jaap Spies (2007-01-25)
Special Functions: __init__,
_eval,
_repr_
self, n) |
sage: [sloane.A000720._eval(n) for n in range(1,11)] [0, 1, 2, 2, 3, 3, 4, 4, 4, 4]
self) |
sage: sloane.A000720._repr_() 'pi(n), the number of primes <= n. Sometimes called PrimePi(n)'
Class: A000796
self) |
Decimal expansion of
.
Input:
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)
Functions: list,
pi
self, n) |
sage: sloane.A000796.list(10) [3, 1, 4, 1, 5, 9, 2, 6, 5, 3]
self) |
Based on a algorithm of Lambert Meertens The ABC-programming language!!!
sage: it = sloane.A000796.pi() sage: [it.next() for i in range(10)] [3, 1, 4, 1, 5, 9, 2, 6, 5, 3]
Special Functions: __init__,
_eval,
_precompute,
_repr_
self, n) |
sage: [sloane.A000796._eval(n) for n in range(1,11)] [3, 1, 4, 1, 5, 9, 2, 6, 5, 3]
self, [how_many=1000]) |
sage: initial = len(sloane.A000796._b) sage: sloane.A000796._precompute(10) sage: len(sloane.A000796._b) - initial 10
self) |
sage: sloane.A000796._repr_() 'Decimal expansion of Pi.'
Class: A000961
self) |
Prime powers
Input:
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]
Author: Jaap Spies (2007-01-25)
Functions: list
self, n) |
sage: sloane.A000961.list(10) [1, 2, 3, 4, 5, 7, 8, 9, 11, 13]
Special Functions: __init__,
_eval,
_precompute,
_repr_
self, n) |
sage: [sloane.A000961._eval(n) for n in range(1,11)] [1, 2, 3, 4, 5, 7, 8, 9, 11, 13]
self, [how_many=150]) |
sage: initial = len(sloane.A000961._b) sage: sloane.A000961._precompute() sage: len(sloane.A000961._b) - initial > 0 True
self) |
sage: sloane.A000961._repr_() 'Prime powers.'
Class: A000984
self) |
Central binomial coefficients:
.
Input:
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]
Author: Jaap Spies (2007-01-26)
Special Functions: __init__,
_eval,
_repr_
self, n) |
sage: [sloane.A000984._eval(n) for n in range(10)] [1, 2, 6, 20, 70, 252, 924, 3432, 12870, 48620]
self) |
sage: sloane.A000984._repr_() 'Central binomial coefficients: C(2n,n) = (2n)!/(n!)^2'
Class: A001006
self) |
Motzkin numbers: number of ways of drawing any number of nonintersecting chords among
points on a circle.
Input:
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]
Author: Jaap Spies (2007-02-02)
Special Functions: __init__,
_eval,
_repr_
self, n) |
sage: [sloane.A001006._eval(n) for n in range(10)] [1, 1, 2, 4, 9, 21, 51, 127, 323, 835]
self) |
sage: sloane.A001006._repr_() 'Motzkin numbers: number of ways of drawing any number of nonintersecting chords among n points on a circle.'
Class: A001045
self) |
Jacobsthal sequence:
,
and
.
Input:
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]
Author: Jaap Spies (2007-01-26)
Special Functions: __init__,
_repr_
self) |
sage: sloane.A001045._repr_() 'Jacobsthal sequence: a(n) = a(n-1) + 2a(n-2).'
Class: A001055
self) |
Number of ways of factoring
with all factors > 1.
Input:
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]
Author: Jaap Spies (2007-02-04)
Functions: nwf
self, n, m) |
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
Special Functions: __init__,
_eval,
_repr_
self, n) |
sage: [sloane.A001055._eval(n) for n in range(1,11)] [1, 1, 1, 2, 1, 2, 1, 3, 2, 2]
self) |
sage: sloane.A001055._repr_() 'Number of ways of factoring n with all factors >1.'
Class: A001109
self) |
is a triangular number:
with
,
.
Input:
sage: a = sloane.A001109;a a(n)^2 is a triangular number: a(n) = 6*a(n-1) - a(n-2) with a(0)=0, a(1)=1 sage: a(0) 0 sage: a(1) 1 sage: a(2) 6 sage: a.offset 0 sage: a(8) 235416 sage: a(60) 1515330104844857898115857393785728383101709300 sage: a.list(9) [0, 1, 6, 35, 204, 1189, 6930, 40391, 235416]
Author: Jaap Spies (2007-01-24)
Special Functions: __init__,
_repr_
self) |
sage: sloane.A001109._repr_() 'a(n)^2 is a triangular number: a(n) = 6*a(n-1) - a(n-2) with a(0)=0, a(1)=1'
Class: A001110
self) |
Numbers that are both triangular and square:
.
Input:
sage: a = sloane.A001110; a Numbers that are both triangular and square: a(n) = 34a(n-1) - a(n-2) + 2. sage: a(0) 0 sage: a(1) 1 sage: a(8) 55420693056 sage: a(21) 4446390382511295358038307980025 sage: a.list(8) [0, 1, 36, 1225, 41616, 1413721, 48024900, 1631432881]
Author: Jaap Spies (2007-01-19)
Functions: g
self, k) |
sage: sloane.A001110.g(2) 2 sage: sloane.A001110.g(1) 0
Special Functions: __init__,
_repr_
self) |
sage: sloane.A001110._repr_() 'Numbers that are both triangular and square: a(n) = 34a(n-1) - a(n-2) + 2.'
Class: A001147
self) |
Double factorial numbers:
.
Input:
sage: a = sloane.A001147;a Double factorial numbers: (2n-1)!! = 1.3.5....(2n-1). sage: a(0) 1 sage: a.offset 0 sage: a(8) 2027025 sage: a(20) 319830986772877770815625 sage: a.list(9) [1, 1, 3, 15, 105, 945, 10395, 135135, 2027025]
Author: Jaap Spies (2007-01-24)
Special Functions: __init__,
_eval,
_repr_
self, n) |
sage: [sloane.A001147._eval(n) for n in range(10)] [1, 1, 3, 15, 105, 945, 10395, 135135, 2027025, 34459425]
self) |
sage: sloane.A001147._repr_() 'Double factorial numbers: (2n-1)!! = 1.3.5....(2n-1).'
Class: A001157
self) |
The sequence
, sum of squares of divisors of
.
The function sigma(n, k) implements
in SAGE.
Input:
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]
Author: Jaap Spies (2007-01-13)
Special Functions: __init__,
_eval,
_repr_
self, n) |
sage: [sloane.A001157._eval(n) for n in range(1,11)] [1, 5, 10, 21, 26, 50, 50, 85, 91, 130]
self) |
sage: sloane.A001157._repr_() 'sigma_2(n): sum of squares of divisors of n'
Class: A001189
self) |
Number of degree-n permutations of order exactly 2.
Input:
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]
Author: Jaap Spies (2007-02-03)
Special Functions: __init__,
_eval,
_repr_
self, n) |
sage: [sloane.A001189._eval(n) for n in range(1,11)] [0, 1, 3, 9, 25, 75, 231, 763, 2619, 9495]
self) |
sage: sloane.A001189._repr_() 'Number of degree-n permutations of order exactly 2.'
Class: A001221
self) |
Number of different prime divisors of
Also called omega(n) or
.
Maximal number of terms in any factorization of
.
Number of prime powers that divide
.
Input:
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]
Author: - Jaap Spies (2007-01-19)
Special Functions: __init__,
_eval,
_repr_
self, n) |
sage: [sloane.A001221._eval(n) for n in range(1,10)] [0, 1, 1, 1, 1, 2, 1, 1, 1]
self) |
sage: sloane.A001221._repr_() 'Number of distinct primes dividing n (also called omega(n)).'
Class: A001222
self) |
Number of prime divisors of
(counted with multiplicity).
Also called bigomega(n) or
.
Maximal number of terms in any factorization of
.
Number of prime powers that divide
.
Input:
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]
Author: - Jaap Spies (2007-01-19)
Special Functions: __init__,
_eval,
_repr_
self, n) |
sage: [sloane.A001222._eval(n) for n in range(1,10)] [0, 1, 1, 2, 1, 2, 1, 3, 2]
self) |
sage: sloane.A001222._repr_() 'Number of prime divisors of n (counted with multiplicity).'
Class: A001227
self) |
Number of odd divisors of
.
Input:
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
Author: - Jaap Spies (2007-01-14)
Special Functions: __init__,
_eval,
_repr_
self, n) |
sage: [sloane.A001227._eval(n) for n in range(1,10)] [1, 1, 2, 1, 2, 2, 2, 1, 3]
self) |
sage: sloane.A001227._repr_() 'Number of odd divisors of n'
Class: A001333
self) |
Numerators of continued fraction convergents to
.
See also A000129
Input:
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]
Author: Jaap Spies (2007-02-01)
Special Functions: __init__,
_repr_
self) |
sage: sloane.A001333._repr_() 'Numerators of continued fraction convergents to sqrt(2).'
Class: A001358
self) |
Products of two primes.
These numbers have been called semiprimes (or semi-primes), biprimes or 2-almost primes.
Input:
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]
Author: Jaap Spies (2007-01-25)
Functions: list
self, n) |
sage: sloane.A001358.list(9) [4, 6, 9, 10, 14, 15, 21, 22, 25]
Special Functions: __init__,
_eval,
_precompute,
_repr_
self, n) |
sage: [sloane.A001358._eval(n) for n in range(1,10)] [4, 6, 9, 10, 14, 15, 21, 22, 25]
self, [how_many=150]) |
sage: initial = len(sloane.A001358._b) sage: sloane.A001358._precompute() sage: len(sloane.A001358._b) - initial > 0 True
self) |
sage: sloane.A001358._repr_() 'Products of two primes.'
Class: A001405
self) |
Central binomial coefficients:
.
Input:
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]
Author: Jaap Spies (2007-01-26)
Special Functions: __init__,
_eval,
_repr_
self, n) |
sage: [sloane.A001405._eval(n) for n in range(10)] [1, 1, 2, 3, 6, 10, 20, 35, 70, 126]
self) |
sage: sloane.A001405._repr_() 'Central binomial coefficients: C(n,floor(n/2)).'
Class: A001477
self) |
The nonnegative integers.
Input:
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]
Author: Jaap Spies (2007-01-26)
Special Functions: __init__,
_eval,
_repr_
self, n) |
sage: [sloane.A001477._eval(n) for n in range(10)] [0, 1, 2, 3, 4, 5, 6, 7, 8, 9]
self) |
sage: sloane.A001477._repr_() 'The nonnegative integers.'
Class: A001694
self) |
This function returns the
-th Powerful Number:
A positive integer
is powerful if for every prime
dividing
,
also divides
.
Input:
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
Author: Jaap Spies (2007-01-14)
Functions: is_powerful,
list
self, n) |
This function returns True if and only if
is a Powerful Number:
A positive integer
is powerful if for every prime
dividing
,
also divides
.
See Sloane's OEIS A001694.
Input:
sage: a = sloane.A001694 sage: a.is_powerful(2500) True sage: a.is_powerful(20) False
Author: - Jaap Spies (2006-12-07)
self, n) |
sage: sloane.A001694.list(9) [1, 4, 8, 9, 16, 25, 27, 32, 36]
Special Functions: __init__,
_eval,
_powerful_numbers_in_range,
_precompute,
_repr_
self, n) |
sage: [sloane.A001694._eval(n) for n in range(1,10)] [1, 4, 8, 9, 16, 25, 27, 32, 36]
self, n, m) |
sage: sloane.A001694._powerful_numbers_in_range(0,50) [4, 8, 9, 16, 25, 27, 32, 36, 49]
self, [how_many=10000]) |
sage: initial = len(sloane.A001694._b) sage: sloane.A001694._precompute() sage: len(sloane.A001694._b) - initial > 0 True
self) |
sage: sloane.A001694._repr_() 'Powerful Numbers (also called squarefull, square-full or 2-full numbers).'
Class: A001836
self) |
Numbers
such that
, where
is
Euler's totient function.
Eulers totient function is also known as euler_phi, euler_phi is a standard SAGE function.
Input:
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 Eler's totient function A000010. [53, 83, 158, 263, 293, 368, 578, 683, 743, 788, 878, 893]
Author: Jaap Spies (2007-01-17)
Functions: list
self, n) |
sage: sloane.A001836.list(9) [53, 83, 158, 263, 293, 368, 578, 683, 743]
Special Functions: __init__,
_eval,
_precompute,
_repr_
self, n) |
sage: [sloane.A001836._eval(n) for n in range(1,10)] [53, 83, 158, 263, 293, 368, 578, 683, 743]
self, [how_many=150]) |
sage: initial = len(sloane.A001836._b) sage: sloane.A001836._precompute() sage: len(sloane.A001836._b) - initial > 0 True
self) |
sage: sloane.A001836._repr_() "Numbers n such that phi(2n-1) < phi(2n), where phi is Euler's totient function A000010."
Class: A001906
self) |
bisection of Fibonacci sequence:
.
Input:
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]
Author: Jaap Spies (2007-01-19)
Special Functions: __init__,
_repr_
self) |
sage: sloane.A001906._repr_() 'F(2n) = bisection of Fibonacci sequence: a(n)=3a(n-1)-a(n-2).'
Class: A001909
self) |
, with
,
.
With offset 1, permanent of (0,1)-matrix of size
with
and
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:
sage: a = sloane.A001909;a a(n) = n*a(n-1) + (n-4)*a(n-2), a(2) = 0, a(3) = 1. sage: a(1) Traceback (most recent call last): ... ValueError: input n (=1) must be an integer >= 2 sage: a.offset 2 sage: a(2) 0 sage: a(8) 8544 sage: a(22) 470033715095287415734 sage: a.list(9) [0, 1, 4, 21, 134, 1001, 8544, 81901, 870274]
Author: Jaap Spies (2007-01-13)
Special Functions: __init__,
_repr_
self) |
sage: sloane.A001909._repr_() 'a(n) = n*a(n-1) + (n-4)*a(n-2), a(2) = 0, a(3) = 1.'
Class: A001910
self) |
, with
,
.
With offset 1, permanent of (0,1)-matrix of size
with
and
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:
sage: a = sloane.A001910;a a(n) = n*a(n-1) + (n-5)*a(n-2), a(3) = 0, a(4) = 1. sage: a(0) Traceback (most recent call last): ... ValueError: input n (=0) must be an integer >= 3 sage: a(3) 0 sage: a.offset 3 sage: a(8) 1909 sage: a(22) 98125321641110663023 sage: a.list(9) [0, 1, 5, 31, 227, 1909, 18089, 190435, 2203319]
Author: Jaap Spies (2007-01-13)
Special Functions: __init__,
_repr_
self) |
sage: sloane.A001910._repr_() 'a(n) = n*a(n-1) + (n-5)*a(n-2), a(3) = 0, a(4) = 1.'
Class: A001969
self) |
Evil numbers: even number of 1's in binary expansion.
Input:
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]
Author: Jaap Spies (2007-02-02)
Special Functions: __init__,
_eval,
_repr_
self, n) |
sage: [sloane.A001969._eval(n) for n in range(10)] [0, 3, 5, 6, 9, 10, 12, 15, 17, 18]
self) |
sage: sloane.A001969._repr_() "Evil numbers: even number of 1's in binary expansion."
Class: A002110
self) |
Primorial numbers (first definition): product of first
primes. Sometimes written
.
Input:
sage: a = sloane.A002110;a Primorial numbers (first definition): product of first n primes. Sometimes written p#. sage: a(0) 1 sage: a(2) 6 sage: a(8) 9699690 sage: a(17) 1922760350154212639070 sage: a.list(9) [1, 2, 6, 30, 210, 2310, 30030, 510510, 9699690]
Author: Jaap Spies (2007-01-25)
Special Functions: __init__,
_eval,
_repr_
self, n) |
sage: [sloane.A002110._eval(n) for n in range(10)] [1, 2, 6, 30, 210, 2310, 30030, 510510, 9699690, 223092870]
self) |
sage: sloane.A002110._repr_() 'Primorial numbers (first definition): product of first n primes. Sometimes written p#.'
Class: A002113
self) |
Palindromes in base 10.
Input:
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]
Author: Jaap Spies (2007-02-02)
Functions: list
self, n) |
sage: sloane.A002113.list(15) [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 22, 33, 44, 55]
Special Functions: __init__,
_eval,
_precompute,
_repr_
self, n) |
sage: [sloane.A002113._eval(n) for n in range(10)] [0, 1, 2, 3, 4, 5, 6, 7, 8, 9]
self, [how_many=150]) |
sage: initial = len(sloane.A002113._b) sage: sloane.A002113._precompute() sage: len(sloane.A002113._b) - initial > 0 True
self) |
sage: sloane.A002113._repr_() 'Palindromes in base 10.'
Class: A002275
self) |
Repunits:
. Often denoted by
.
Input:
sage: a = sloane.A002275;a Repunits: (10^n - 1)/9. Often denoted by R_n. sage: a(0) 0 sage: a(2) 11 sage: a(8) 11111111 sage: a(20) 11111111111111111111 sage: a.list(9) [0, 1, 11, 111, 1111, 11111, 111111, 1111111, 11111111]
Author: Jaap Spies (2007-01-25)
Special Functions: __init__,
_eval,
_repr_
self, n) |
sage: [sloane.A002275._eval(n) for n in range(10)] [0, 1, 11, 111, 1111, 11111, 111111, 1111111, 11111111, 111111111]
self) |
sage: sloane.A002275._repr_() 'Repunits: (10^n - 1)/9. Often denoted by R_n.'
Class: A002378
self) |
Oblong (or pronic, or heteromecic) numbers:
.
Input:
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]
Author: Jaap Spies (2007-01-26)
Special Functions: __init__,
_eval,
_repr_
self, n) |
sage: [sloane.A002378._eval(n) for n in range(10)] [0, 2, 6, 12, 20, 30, 42, 56, 72, 90]
self) |
sage: sloane.A002378._repr_() 'Oblong (or pronic, or heteromecic) numbers: n(n+1).'
Class: A002620
self) |
Quarter-squares: floor(n/2)*ceiling(n/2). Equivalently,
.
Input:
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]
Author: Jaap Spies (2007-01-26)
Special Functions: __init__,
_eval,
_repr_
self, n) |
sage: [sloane.A002620._eval(n) for n in range(10)] [0, 0, 1, 2, 4, 6, 9, 12, 16, 20]
self) |
sage: sloane.A002620._repr_() 'Quarter-squares: floor(n/2)*ceiling(n/2). Equivalently, floor(n^2/4).'
Class: A002808
self) |
The composite numbers: numbers
of the form
for
and
.
Input:
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]
Author: Jaap Spies (2007-01-26)
Functions: list
self, n) |
sage: sloane.A002808.list(10) [4, 6, 8, 9, 10, 12, 14, 15, 16, 18]
Special Functions: __init__,
_eval,
_precompute,
_repr_
self, n) |
sage: [sloane.A002808._eval(n) for n in range(1,11)] [4, 6, 8, 9, 10, 12, 14, 15, 16, 18]
self, [how_many=150]) |
sage: initial = len(sloane.A002808._b) sage: sloane.A002808._precompute() sage: len(sloane.A002808._b) - initial > 0 True
self) |
sage: sloane.A002808._repr_() 'The composite numbers: numbers n of the form x*y for x > 1 and y > 1.'
Class: A003418
self) |
Least common multiple (or lcm) of
.
Input:
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, long, or Integer
Author: Jaap Spies (2007-01-31)
Special Functions: __init__,
_eval,
_repr_
self, n) |
sage: [sloane.A003418._eval(n) for n in range(1,11)] [1, 2, 6, 12, 60, 60, 420, 840, 2520, 2520]
self) |
sage: sloane.A003418._repr_() 'Least common multiple (or lcm) of {1, 2, ..., n}.'
Class: A004086
self) |
Read n backwards (referred to as
in many sequences).
Input:
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]
Author: Jaap Spies (2007-02-02)
Special Functions: __init__,
_eval,
_repr_
self, n) |
sage: [sloane.A004086._eval(n) for n in range(10)] [0, 1, 2, 3, 4, 5, 6, 7, 8, 9]
self) |
sage: sloane.A004086._repr_() 'Read n backwards (referred to as R(n) in many sequences).'
Class: A004526
self) |
The nonnegative integers repeated`
Input:
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]
Author: Jaap Spies (2007-01-26)
Special Functions: __init__,
_eval,
_repr_
self, n) |
sage: [sloane.A004526._eval(n) for n in range(10)] [0, 0, 1, 1, 2, 2, 3, 3, 4, 4]
self) |
sage: sloane.A004526._repr_() 'The nonnegative integers repeated.'
Class: A005100
self) |
Deficient numbers:
.
Input:
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]
Author: Jaap Spies (2007-01-26)
Functions: list
self, n) |
sage: sloane.A005100.list(10) [1, 2, 3, 4, 5, 7, 8, 9, 10, 11]
Special Functions: __init__,
_eval,
_precompute,
_repr_
self, n) |
sage: [sloane.A005100._eval(n) for n in range(1,10)] [1, 2, 3, 4, 5, 7, 8, 9, 10]
self, [how_many=150]) |
sage: initial = len(sloane.A005100._b) sage: sloane.A005100._precompute() sage: len(sloane.A005100._b) - initial > 0 True
self) |
sage: sloane.A005100._repr_() 'Deficient numbers: sigma(n) < 2n'
Class: A005101
self) |
Abundant numbers (sum of divisors of
exceeds
).
Input:
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]
Author: Jaap Spies (2007-01-26)
Functions: list
self, n) |
sage: sloane.A005101.list(10) [12, 18, 20, 24, 30, 36, 40, 42, 48, 54]
Special Functions: __init__,
_eval,
_precompute,
_repr_
self, n) |
sage: [sloane.A005101._eval(n) for n in range(1,11)] [12, 18, 20, 24, 30, 36, 40, 42, 48, 54]
self, [how_many=150]) |
sage: initial = len(sloane.A005101._b) sage: sloane.A005101._precompute() sage: len(sloane.A005101._b) - initial > 0 True
self) |
sage: sloane.A005101._repr_() 'Abundant numbers (sum of divisors of n exceeds 2n).'
Class: A005117
self) |
Square-free numbers
Input:
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]
Author: Jaap Spies (2007-01-25)
Functions: list
self, n) |
sage: sloane.A005117.list(10) [1, 2, 3, 5, 6, 7, 10, 11, 13, 14]
Special Functions: __init__,
_eval,
_precompute,
_repr_
self, n) |
sage: [sloane.A005117._eval(n) for n in range(1,11)] [1, 2, 3, 5, 6, 7, 10, 11, 13, 14]
self, [how_many=150]) |
sage: initial = len(sloane.A005117._b) sage: sloane.A005117._precompute() sage: len(sloane.A005117._b) - initial > 0 True
self) |
sage: sloane.A005117._repr_() 'Square-free numbers.'
Class: A005408
self) |
The odd numbers a(n) = 2n + 1.
Input:
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]
Author: Jaap Spies (2007-01-26)
Special Functions: __init__,
_eval,
_repr_
self, n) |
sage: [sloane.A005408._eval(n) for n in range(10)] [1, 3, 5, 7, 9, 11, 13, 15, 17, 19]
self) |
sage: sloane.A005408._repr_() 'The odd numbers a(n) = 2n + 1.'
Class: A005843
self) |
The even numbers:
.
Input:
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, long, 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]
Author: Jaap Spies (2007-02-03)
Special Functions: __init__,
_eval,
_repr_
self, n) |
sage: [sloane.A005843._eval(n) for n in range(10)] [0, 2, 4, 6, 8, 10, 12, 14, 16, 18]
self) |
sage: sloane.A005843._repr_() 'The even numbers: a(n) = 2n.'
Class: A006318
self) |
Large Schroeder numbers.
Input:
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]
Author: Jaap Spies (2007-02-03)
Special Functions: __init__,
_eval,
_repr_
self, n) |
sage: [sloane.A006318._eval(n) for n in range(10)] [1, 2, 6, 22, 90, 394, 1806, 8558, 41586, 206098]
self) |
sage: sloane.A006318._repr_() 'Large Schroeder numbers.'
Class: A006530
self) |
Largest prime dividing
(with
).
Input:
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]
Author: Jaap Spies (2007-01-25)
Special Functions: __init__,
_eval,
_repr_
self, n) |
sage: [sloane.A006530._eval(n) for n in range(1,11)] [1, 2, 3, 2, 5, 3, 7, 2, 3, 5]
self) |
sage: sloane.A006530._repr_() 'Largest prime dividing n (with a(1)=1).'
Class: A006882
self) |
Double factorials
:
.
Input:
sage: a = sloane.A006882;a Double factorials n!!: a(n)=n*a(n-2). sage: a(0) 1 sage: a(2) 2 sage: a(8) 384 sage: a(20) 3715891200 sage: a.list(9) [1, 1, 2, 3, 8, 15, 48, 105, 384]
Author: Jaap Spies (2007-01-24)
Functions: df,
list
self) |
Double factorials n!!: a(n)=n*a(n-2).
sage: it = sloane.A006882.df() sage: [it.next() for i in range(10)] [1, 1, 2, 3, 8, 15, 48, 105, 384, 945]
self, n) |
sage: sloane.A006882.list(10) [1, 1, 2, 3, 8, 15, 48, 105, 384, 945]
Special Functions: __init__,
_eval,
_precompute,
_repr_
self, n) |
sage: [sloane.A006882._eval(n) for n in range(10)] [1, 1, 2, 3, 8, 15, 48, 105, 384, 945]
self, [how_many=10]) |
sage: initial = len(sloane.A006882._b) sage: sloane.A006882._precompute(10) sage: len(sloane.A006882._b) - initial == 10 True
self) |
sage: sloane.A006882._repr_() 'Double factorials n!!: a(n)=n*a(n-2).'
Class: A007318
self) |
Pascal's triangle read by rows:
,
.
Input:
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
Author: Jaap Spies (2007-01-31)
Special Functions: __init__,
_eval,
_repr_
self, n) |
sage: [sloane.A007318._eval(n) for n in range(10)] [1, 1, 1, 1, 2, 1, 1, 3, 3, 1]
self) |
sage: sloane.A007318._repr_() "Pascal's triangle read by rows: C(n,k) = binomial(n,k) = n!/(k!*(n-k)!), 0<=k<=n."
Class: A008275
self) |
Triangle of Stirling numbers of first kind,
,
,
.
The unsigned numbers are also called Stirling cycle numbers:
= number of permutations of
objects with exactly
cycles.
Input:
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]
Author: Jaap Spies (2007-02-02)
Functions: s
self, n, k) |
sage: sloane.A008275.s(4,2) 11 sage: sloane.A008275.s(5,2) -50 sage: sloane.A008275.s(5,3) 35
Special Functions: __init__,
_eval,
_repr_
self, n) |
sage: [sloane.A008275._eval(n) for n in range(1, 11)] [1, -1, 1, 2, -3, 1, -6, 11, -6, 1]
self) |
sage: sloane.A008275._repr_() 'Triangle of Stirling numbers of first kind, s(n,k), n >= 1, 1<=k<=n.'
Class: A008277
self) |
Triangle of Stirling numbers of 2nd kind,
,
,
.
Input:
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, long, or Integer sage: a.list(15) [1, 1, 1, 1, 3, 1, 1, 7, 6, 1, 1, 15, 25, 10, 1]
Author: Jaap Spies (2007-01-31)
Functions: s2
self, n, k) |
Returns the Stirling number S2(n,k) of the 2nd kind.
sage: sloane.A008277.s2(4,2) 7
Special Functions: __init__,
_eval,
_repr_
self, n) |
sage: [sloane.A008277._eval(n) for n in range(1,11)] [1, 1, 1, 1, 3, 1, 1, 7, 6, 1]
self) |
sage: sloane.A008277._repr_() 'Triangle of Stirling numbers of 2nd kind, S2(n,k), n >= 1, 1<=k<=n.'
Class: A008683
self) |
Moebius (or Möbius) function
.
Input:
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]
Author: Jaap Spies (2007-01-13)
Special Functions: __init__,
_eval,
_repr_
self, n) |
sage: [sloane.A008683._eval(n) for n in range(1,11)] [1, -1, -1, 0, -1, 1, -1, 0, 0, 1]
self) |
sage: sloane.A008683._repr_() 'Moebius function mu(n).'
Class: A010060
self) |
Thue-Morse sequence.
Let
denote the first
terms; then
, and for
,
,
where
is obtained from
by interchanging 0's and 1's.
Input:
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]
Author: Jaap Spies (2007-02-02)
Special Functions: __init__,
_eval,
_repr_
self, n) |
sage: [sloane.A010060._eval(n) for n in range(10)] [0, 1, 1, 0, 1, 0, 0, 1, 1, 0]
self) |
sage: sloane.A010060._repr_() 'Thue-Morse sequence.'
Class: A015521
self) |
Linear 2nd order recurrence,
,
and
.
Input:
sage: a = sloane.A015521; a Linear 2nd order recurrence, a(n) = 3 a(n-1) + 4 a(n-2). sage: a(0) 0 sage: a(1) 1 sage: a(8) 13107 sage: a(41) 967140655691703339764941 sage: a.list(12) [0, 1, 3, 13, 51, 205, 819, 3277, 13107, 52429, 209715, 838861]
Author: Jaap Spies (2007-01-19)
Special Functions: __init__,
_repr_
self) |
sage: sloane.A015521._repr_() 'Linear 2nd order recurrence, a(n) = 3 a(n-1) + 4 a(n-2).'
Class: A015523
self) |
Linear 2nd order recurrence,
,
and
.
Input:
sage: a = sloane.A015523; a Linear 2nd order recurrence, a(n) = 3 a(n-1) + 5 a(n-2). sage: a(0) 0 sage: a(1) 1 sage: a(8) 17727 sage: a(41) 6173719566474529739091481 sage: a.list(12) [0, 1, 3, 14, 57, 241, 1008, 4229, 17727, 74326, 311613, 1306469]
Author: Jaap Spies (2007-01-19)
Special Functions: __init__,
_repr_
self) |
sage: sloane.A015523._repr_() 'Linear 2nd order recurrence, a(n) = 3 a(n-1) + 5 a(n-2).'
Class: A015530
self) |
Linear 2nd order recurrence,
,
and
.
Input:
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]
Author: Jaap Spies (2007-01-19)
Special Functions: __init__,
_repr_
self) |
sage: sloane.A015530._repr_() 'Linear 2nd order recurrence, a(n) = 4 a(n-1) + 3 a(n-2).'
Class: A015531
self) |
Linear 2nd order recurrence,
,
and
.
Input:
sage: a = sloane.A015531;a Linear 2nd order recurrence, a(n) = 4 a(n-1) + 5 a(n-2). sage: a(0) 0 sage: a(1) 1 sage: a(2) 4 sage: a.offset 0 sage: a(8) 65104 sage: a(60) 144560289664733924534327040115992228190104 sage: a.list(9) [0, 1, 4, 21, 104, 521, 2604, 13021, 65104]
Author: Jaap Spies (2007-01-19)
Special Functions: __init__,
_repr_
self) |
sage: sloane.A015531._repr_() 'Linear 2nd order recurrence, a(n) = 4 a(n-1) + 5 a(n-2).'
Class: A015551
self) |
Linear 2nd order recurrence,
,
and
.
Input:
sage: a = sloane.A015551;a Linear 2nd order recurrence, a(n) = 6 a(n-1) + 5 a(n-2). sage: a(0) 0 sage: a(1) 1 sage: a(2) 6 sage: a.offset 0 sage: a(8) 570216 sage: a(60) 7110606606530059736761484557155863822531970573036 sage: a.list(9) [0, 1, 6, 41, 276, 1861, 12546, 84581, 570216]
Author: Jaap Spies (2007-01-19)
Special Functions: __init__,
_repr_
self) |
sage: sloane.A015551._repr_() 'Linear 2nd order recurrence, a(n) = 6 a(n-1) + 5 a(n-2).'
Class: A018252
self) |
The nonprime numbers, starting with 1.
Input:
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]
Author: Jaap Spies (2007-02-04)
Special Functions: __init__,
_eval,
_repr_
self, n) |
sage: [sloane.A018252._eval(n) for n in range(1,11)] [1, 4, 6, 8, 9, 10, 12, 14, 15, 16]
self) |
sage: sloane.A018252._repr_() 'The nonprime numbers.'
Class: A020639
self) |
Least prime dividing
with
.
Input:
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]
Author: Jaap Spies (2007-01-25)
Functions: list
self, n) |
sage: sloane.A020639.list(10) [1, 2, 3, 2, 5, 2, 7, 2, 3, 2]
Special Functions: __init__,
_eval,
_precompute,
_repr_
self, n) |
sage: [sloane.A020639._eval(n) for n in range(1,11)] [1, 2, 3, 2, 5, 2, 7, 2, 3, 2]
self, [how_many=50]) |
sage: initial = len(sloane.A020639._b) sage: sloane.A020639._precompute(10) sage: len(sloane.A020639._b) - initial == 10 True
self) |
sage: sloane.A020639._repr_() 'Least prime dividing n (a(1)=1).'
Class: A046660
.
Input:
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]
Author: - Jaap Spies (2007-01-19)
Special Functions: _eval,
_repr_
self, n) |
sage: [sloane.A046660._eval(n) for n in range(1,10)] [0, 0, 0, 1, 0, 0, 0, 2, 1]
self) |
sage: sloane.A046660._repr_() 'Excess of n = Bigomega (with multiplicity) - omega (without multiplicity).'
Class: A049310
self) |
Triangle of coefficients of Chebyshev's
:
polynomials (exponents in increasing order).
Input:
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']
Author: Jaap Spies (2007-01-31)
Special Functions: __init__,
_eval,
_repr_
self, n) |
sage: [sloane.A049310._eval(n) for n in range(10)] [1, 0, 1, -1, 0, 1, 0, -2, 0, 1]
self) |
sage: sloane.A049310._repr_() "Triangle of coefficients of Chebyshev's S(n,x) := U(n,x/2) polynomials (exponents in increasing order)."
Class: A051959
self) |
Linear second order recurrence. A051959.
Input:
sage: a = sloane.A051959; a Linear second order recurrence. A051959. sage: a(0) 1 sage: a(1) 10 sage: a(8) 9969 sage: a(41) 42834431872413650 sage: a.list(12) [1, 10, 36, 104, 273, 686, 1688, 4112, 9969, 24114, 58268, 140728]
Author: Jaap Spies (2007-01-19)
Functions: g
self, k) |
sage: sloane.A051959.g(2) 15 sage: sloane.A051959.g(1) 0
Special Functions: __init__,
_repr_
self) |
sage: sloane.A051959._repr_() 'Linear second order recurrence. A051959.'
Class: A055790
self) |
.
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:
sage: a = sloane.A055790;a a(n) = n*a(n-1) + (n-2)*a(n-2) [a(0) = 0, a(1) = 2]. sage: a(0) 0 sage: a(1) 2 sage: a(2) 4 sage: a.offset 0 sage: a(8) 165016 sage: a(22) 10356214297533070441564 sage: a.list(9) [0, 2, 4, 14, 64, 362, 2428, 18806, 165016]
Author: Jaap Spies (2007-01-23)
Special Functions: __init__,
_repr_
self) |
sage: sloane.A055790._repr_() 'a(n) = n*a(n-1) + (n-2)*a(n-2) [a(0) = 0, a(1) = 2].'
Class: A061084
self) |
Fibonacci-type sequence based on subtraction:
,
and
.
Input:
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']
Author: Jaap Spies (2007-01-18)
Special Functions: __init__,
_eval,
_repr_
self, n) |
sage: [sloane.A061084._eval(n) for n in range(10)] [1, 2, -1, 3, -4, 7, -11, 18, -29, 47]
self) |
sage: sloane.A061084._repr_() 'Fibonacci-type sequence based on subtraction: a(0) = 1, a(1) = 2 and a(n) = a(n-2)-a(n-1).'
Class: A064553
self) |
,
for
and
for
.
Input:
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]
Author: Jaap Spies (2007-02-04)
Special Functions: __init__,
_eval,
_repr_
self, n) |
sage: [sloane.A064553._eval(n) for n in range(1,11)] [1, 2, 3, 4, 4, 6, 5, 8, 9, 8]
self) |
sage: sloane.A064553._repr_() 'a(1) = 1, a(prime(i)) = i+1 for i > 0 and a(u*v) = a(u)*a(v) for u,v > 0'
Class: A079922
The value is
, the permanent of the (0,1)-matrix
of size
with
if and only if
.
REFERENCES: Jaap Spies, Nieuw Archief voor Wiskunde, 5/7 nr 4, December 2006
Input:
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
Author: - Jaap Spies (2007-01-14)
Special Functions: _eval,
_repr_
self, n) |
sage: [sloane.A079922._eval(n) for n in range(1,5)] [4, 13, 36, 90]
self) |
sage: sloane.A079922._repr_() 'Solutions to the Dancing School problem with n girls and n+3 boys'
Class: A079923
The value is
, the permanent of the (0,1)-matrix
of size
with
if and only if
.
REFERENCES: Jaap Spies, Nieuw Archief voor Wiskunde, 5/7 nr 4, December 2006
Input:
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
Author: - Jaap Spies (2007-01-17)
Special Functions: _eval,
_repr_
self, n) |
sage: [sloane.A079923._eval(n) for n in range(1,11)] [5, 21, 76, 246, 738, 2108, 5794, 15458, 40296, 103129]
self) |
sage: sloane.A079923._repr_() 'Solutions to the Dancing School problem with n girls and n+4 boys'
Class: A082411
self) |
Second-order linear recurrence sequence with
.
,
. This is the second-order linear
recurrence sequence with
and
co- prime, that R. L. Graham in 1964
stated did not contain any primes.
Input:
sage: a = sloane.A082411;a Second-order linear recurrence sequence with a(n) = a(n-1) + a(n-2). sage: a(1) 76343678551 sage: a(2) 483732902969 sage: a(3) 560076581520 sage: a(20) 2219759332689173 sage: a.list(4) [407389224418, 76343678551, 483732902969, 560076581520]
Author: Jaap Spies (2007-01-23)
Special Functions: __init__,
_repr_
self) |
sage: sloane.A082411._repr_() 'Second-order linear recurrence sequence with a(n) = a(n-1) + a(n-2).'
Class: A083103
self) |
Second-order linear recurrence sequence with
.
,
. This is the second-order linear
recurrence sequence with
and
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:
sage: a = sloane.A083103;a Second-order linear recurrence sequence with a(n) = a(n-1) + a(n-2). sage: a(1) 1059683225053915111058165141686995 sage: a(2) 2846455926982717743326880272142788 sage: a(3) 3906139152036632854385045413829783 sage: a.offset 0 sage: a(8) 45481392851206651551714764671352204 sage: a(20) 14639253684254059531823985143948191708 sage: a.list(4) [1786772701928802632268715130455793, 1059683225053915111058165141686995, 2846455926982717743326880272142788, 3906139152036632854385045413829783]
Author: Jaap Spies (2007-01-23)
Special Functions: __init__,
_repr_
self) |
sage: sloane.A083103._repr_() 'Second-order linear recurrence sequence with a(n) = a(n-1) + a(n-2).'
Class: A083104
self) |
Second-order linear recurrence sequence with
.
,
. This is the second-order linear
recurrence sequence with
and
co-prime.
It was found by Ronald Graham in 1990.
Input:
sage: a = sloane.A083104;a Second-order linear recurrence sequence with a(n) = a(n-1) + a(n-2). sage: a(3) 3351693458175078679851381267428333 sage: a.offset 0 sage: a(8) 36021870400834012982120004949074404 sage: a(20) 11601914177621826012468849361236300628
Author: Jaap Spies (2007-01-23)
Special Functions: __init__,
_repr_
self) |
sage: sloane.A083104._repr_() 'Second-order linear recurrence sequence with a(n) = a(n-1) + a(n-2).'
Class: A083105
self) |
Second-order linear recurrence sequence with
.
,
. This is the second-order linear
recurrence sequence with
and
co-prime.
It was found by Donald Knuth in 1990.
Input:
sage: a = sloane.A083105;a Second-order linear recurrence sequence with a(n) = a(n-1) + a(n-2). sage: a(1) 49463435743205655 sage: a(2) 112101715747445512 sage: a(3) 161565151490651167 sage: a.offset 0 sage: a(8) 1853029790662436896 sage: a(20) 596510791500513098192 sage: a.list(4) [62638280004239857, 49463435743205655, 112101715747445512, 161565151490651167]
Author: Jaap Spies (2007-01-23)
Special Functions: __init__,
_repr_
self) |
sage: sloane.A083105._repr_() 'Second-order linear recurrence sequence with a(n) = a(n-1) + a(n-2).'
Class: A083216
self) |
Second-order linear recurrence sequence with
.
,
. This is a
second-order linear recurrence sequence with
and
co-prime that does not contain any primes. It was found by Herbert Wilf in 1990.
Input:
sage: a = sloane.A083216; a Second-order linear recurrence sequence with a(n) = a(n-1) + a(n-2). sage: a(0) 20615674205555510 sage: a(1) 3794765361567513 sage: a(8) 347693837265139403 sage: a(41) 2738025383211084205003383 sage: a.list(4) [20615674205555510, 3794765361567513, 24410439567123023, 28205204928690536]
Author: Jaap Spies (2007-01-19)
Special Functions: __init__,
_repr_
self) |
sage: sloane.A083216._repr_() 'Second-order linear recurrence sequence with a(n) = a(n-1) + a(n-2).'
Class: A090010
self) |
Permanent of (0,1)-matrix of size
with
and
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:
sage: a = sloane.A090010;a Permanent of (0,1)-matrix of size n X (n+d) with d=6 and n zeros not on a line. sage: a(0) Traceback (most recent call last): ... ValueError: input n (=0) must be a positive integer sage: a(1) 6 sage: a(2) 43 sage: a.offset 1 sage: a(8) 67741129 sage: a(22) 192416593029158989003270143 sage: a.list(9) [6, 43, 356, 3333, 34754, 398959, 4996032, 67741129, 988344062]
Author: Jaap Spies (2007-01-23)
Special Functions: __init__,
_repr_
self) |
sage: sloane.A090010._repr_() 'Permanent of (0,1)-matrix of size n X (n+d) with d=6 and n zeros not on a line.'
Class: A090012
self) |
Permanent of (0,1)-matrix of size
with
and
zeros not on a line.
,
and
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:
sage: a = sloane.A090012;a Permanent of (0,1)-matrix of size n X (n+d) with d=2 and n-1 zeros not on a line. sage: a(0) Traceback (most recent call last): ... ValueError: input n (=0) must be a positive integer sage: a(1) 3 sage: a(2) 9 sage: a.offset 1 sage: a(8) 890901 sage: a(22) 129020386652297208795129 sage: a.list(9) [3, 9, 39, 213, 1395, 10617, 91911, 890901, 9552387]
Author: Jaap Spies (2007-01-23)
Special Functions: __init__,
_eval,
_repr_
self, n) |
sage: [sloane.A090012._eval(n) for n in range(1,11)] [3, 9, 39, 213, 1395, 10617, 91911, 890901, 9552387, 112203465]
self) |
sage: sloane.A090012._repr_() 'Permanent of (0,1)-matrix of size n X (n+d) with d=2 and n-1 zeros not on a line.'
Class: A090013
self) |
Permanent of (0,1)-matrix of size
with
and
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:
sage: a = sloane.A090013;a Permanent of (0,1)-matrix of size n X (n+d) with d=3 and n-1 zeros not on a line. sage: a(0) Traceback (most recent call last): ... ValueError: input n (=0) must be a positive integer sage: a(1) 4 sage: a(2) 16 sage: a.offset 1 sage: a(8) 3481096 sage: a(22) 1112998577171142607670336 sage: a.list(9) [4, 16, 84, 536, 4004, 34176, 327604, 3481096, 40585284]
Author: Jaap Spies (2007-01-23)
Special Functions: __init__,
_eval,
_repr_
self, n) |
sage: [sloane.A090013._eval(n) for n in range(1,11)] [4, 16, 84, 536, 4004, 34176, 327604, 3481096, 40585284, 514872176]
self) |
sage: sloane.A090013._repr_() 'Permanent of (0,1)-matrix of size n X (n+d) with d=3 and n-1 zeros not on a line.'
Class: A090014
self) |
Permanent of (0,1)-matrix of size
with
and
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:
sage: a = sloane.A090014;a Permanent of (0,1)-matrix of size n X (n+d) with d=4 and n-1 zeros not on a line. sage: a(0) Traceback (most recent call last): ... ValueError: input n (=0) must be a positive integer sage: a(1) 5 sage: a(2) 25 sage: a.offset 1 sage: a(8) 11016595 sage: a(22) 7469733600354446865509725 sage: a.list(9) [5, 25, 155, 1135, 9545, 90445, 952175, 11016595, 138864365]
Author: Jaap Spies (2007-01-23)
Special Functions: __init__,
_eval,
_repr_
self, n) |
sage: [sloane.A090014._eval(n) for n in range(1,11)] [5, 25, 155, 1135, 9545, 90445, 952175, 11016595, 138864365, 1893369505]
self) |
sage: sloane.A090014._repr_() 'Permanent of (0,1)-matrix of size n X (n+d) with d=4 and n-1 zeros not on a line.'
Class: A090015
self) |
Permanent of (0,1)-matrix of size
with
and
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:
sage: a = sloane.A090015;a Permanent of (0,1)-matrix of size n X (n+d) with d=3 and n-1 zeros not on a line. sage: a(0) Traceback (most recent call last): ... ValueError: input n (=0) must be a positive integer sage: a(1) 6 sage: a(2) 36 sage: a.offset 1 sage: a(8) 29976192 sage: a(22) 41552258517692116794936876 sage: a.list(9) [6, 36, 258, 2136, 19998, 208524, 2393754, 29976192, 406446774]
Author: Jaap Spies (2007-01-23)
Special Functions: __init__,
_eval,
_repr_
self, n) |
sage: [sloane.A090015._eval(n) for n in range(1,10)] [6, 36, 258, 2136, 19998, 208524, 2393754, 29976192, 406446774]
self) |
sage: sloane.A090015._repr_() 'Permanent of (0,1)-matrix of size n X (n+d) with d=3 and n-1 zeros not on a line.'
Class: A090016
self) |
Permanent of (0,1)-matrix of size
with
and
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:
sage: a = sloane.A090016;a Permanent of (0,1)-matrix of size n X (n+d) with d=6 and n-1 zeros not on a line. sage: a(0) Traceback (most recent call last): ... ValueError: input n (=0) must be a positive integer sage: a(1) 7 sage: a(2) 49 sage: a.offset 1 sage: a(8) 72737161 sage: a(22) 199341969448774341802426289 sage: a.list(9) [7, 49, 399, 3689, 38087, 433713, 5394991, 72737161, 1056085191]
Author: Jaap Spies (2007-01-23)
Special Functions: __init__,
_eval,
_repr_
self, n) |
sage: [sloane.A090016._eval(n) for n in range(1,10)] [7, 49, 399, 3689, 38087, 433713, 5394991, 72737161, 1056085191]
self) |
sage: sloane.A090016._repr_() 'Permanent of (0,1)-matrix of size n X (n+d) with d=6 and n-1 zeros not on a line.'
Class: A111774
self) |
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
consecutive
integers (other than the trivial
for
).
See: http://www.jaapspies.nl/mathfiles/problem2005-2C.pdf
Input:
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, long, or Integer
Author: Jaap Spies (2007-01-13)
Functions: is_number_of_the_third_kind,
list
self, n) |
This function returns True if and only if
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
consecutive integers (other
than the trivial
for
).
See: http://www.jaapspies.nl/mathfiles/problem2005-2C.pdf
Input:
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
Author: Jaap Spies (2006-12-09)
self, n) |
sage: sloane.A111774.list(12) [6, 9, 10, 12, 14, 15, 18, 20, 21, 22, 24, 25]
Special Functions: __init__,
_eval,
_precompute,
_repr_
self, n) |
sage: [sloane.A111774._eval(n) for n in range(1,11)] [6, 9, 10, 12, 14, 15, 18, 20, 21, 22]
self, [how_many=150]) |
sage: initial = len(sloane.A111774._b) sage: sloane.A111774._precompute() sage: len(sloane.A111774._b) - initial > 0 True
self) |
sage: sloane.A111774._repr_() 'Numbers that can be written as a sum of at least three consecutive positive integers.'
Class: A111775
self) |
Number of ways
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.
strongly depends on the number
of odd divisors of
(A001227):
Suppose
is to be written as sum of
consecutive integers
starting with
, then
.
Only one of the factors is odd. For each odd divisor of
there is a unique corresponding
,
and
must be excluded.
See: http://www.jaapspies.nl/mathfiles/problem2005-2C.pdf
Input:
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, long, or Integer
Author: Jaap Spies (2006-12-09)
Special Functions: __init__,
_eval,
_repr_
self, n) |
sage: [sloane.A111775._eval(n) for n in range(10)] [0, 0, 0, 0, 0, 0, 1, 0, 0, 1]
self) |
sage: sloane.A111775._repr_() 'Number of ways n can be written as a sum of at least three consecutive integers.'
Class: A111776
self) |
The
th term of the sequence
is the largest
such that
can be written as sum of
consecutive integers.
is the sum of at most
consecutive positive integers.
Suppose
is to be written as sum of
consecutive integers starting
with
, then
. Only one of the factors is odd.
For each odd divisor
of
there is a unique corresponding
.
is the largest among those
.
See: http://www.jaapspies.nl/mathfiles/problem2005-2C.pdf
Input:
sage: a = sloane.A111776; a a(n) is the largest k such that n can be written as sum of k consecutive integers. sage: a(0) 1 sage: a(2) 1 sage: a.list(9) [1, 1, 1, 2, 1, 2, 3, 2, 1]
Author: Jaap Spies (2007-01-13)
Special Functions: __init__,
_eval,
_repr_
self, n) |
sage: [sloane.A111776._eval(n) for n in range(10)] [1, 1, 1, 2, 1, 2, 3, 2, 1, 3]
self) |
sage: sloane.A111776._repr_() 'a(n) is the largest k such that n can be written as sum of k consecutive integers.'
Class: A111787
self) |
This function returns the
-th number of Sloane's sequence A111787
if
is an odd prime or a power of 2. For numbers of the third
kind (see A111774) we proceed as follows: suppose
is to be written as sum of
consecutive integers starting with
, then
.
Let
be the smallest odd prime divisor of
then
.
See: http://www.jaapspies.nl/mathfiles/problem2005-2C.pdf
Input:
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
Author: - Jaap Spies (2007-01-14)
Special Functions: __init__,
_eval,
_repr_
self, n) |
sage: [sloane.A111787._eval(n) for n in range(1,11)] [0, 0, 0, 0, 0, 3, 0, 0, 3, 4]
self) |
sage: sloane.A111787._repr_() '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.'
Class: ExponentialNumbers
self, a) |
A sequence of Exponential numbers.
sage: from sage.combinat.sloane_functions import ExponentialNumbers sage: ExponentialNumbers(0) Sequence of Exponential numbers around 0
Special Functions: __init__,
_eval,
_repr_
self, n) |
sage: [sloane.A000110._eval(n) for n in range(10)] [1, 1, 2, 5, 15, 52, 203, 877, 4140, 21147]
self) |
sage: from sage.combinat.sloane_functions import ExponentialNumbers sage: ExponentialNumbers(4)._repr_() 'Sequence of Exponential numbers around 4'
Class: ExtremesOfPermanentsSequence
Functions: gen,
list
self, a0, a1, d) |
sage: it = sloane.A000153.gen(0,1,2) sage: [it.next() for i in range(5)] [0, 1, 2, 7, 32]
self, n) |
sage: sloane.A000153.list(8) [0, 1, 2, 7, 32, 181, 1214, 9403]
Special Functions: _eval,
_precompute
self, n) |
sage: [sloane.A000153._eval(n) for n in range(8)] [0, 1, 2, 7, 32, 181, 1214, 9403]
self, [how_many=20]) |
sage: sloane.A000153._precompute() sage: v1 = len(sloane.A000153._b) sage: sloane.A000153._precompute(10) sage: len(sloane.A000153._b) - v1 10
Class: ExtremesOfPermanentsSequence2
Functions: gen
self, a0, a1, d) |
sage: from sage.combinat.sloane_functions import ExtremesOfPermanentsSequence2 sage: e = ExtremesOfPermanentsSequence2() sage: it = e.gen(6,43,6) sage: [it.next() for i in range(5)] [6, 43, 307, 2542, 23799]
Class: RecurrenceSequence
Functions: list
self, n) |
sage: sloane.A001110.list(8) [0, 1, 36, 1225, 41616, 1413721, 48024900, 1631432881]
Special Functions: _eval,
_precompute
self, n) |
sage: [sloane.A001110._eval(n) for n in range(5)] [0, 1, 36, 1225, 41616]
self, [how_many=20]) |
sage: initial = len(sloane.A001110._b) sage: sloane.A001110._precompute(10) sage: len(sloane.A001110._b) - initial == 10 True
Class: RecurrenceSequence2
Functions: list
self, n) |
sage: sloane.A001906.list(10) [0, 1, 3, 8, 21, 55, 144, 377, 987, 2584]
Special Functions: _eval,
_precompute
self, n) |
sage: [sloane.A001906._eval(n) for n in range(10)] [0, 1, 3, 8, 21, 55, 144, 377, 987, 2584]
self, [how_many=150]) |
sage: initial = len(sloane.A001906._b) sage: sloane.A001906._precompute(10) sage: len(sloane.A001906._b) - initial == 10 True
Class: Sloane
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.
Ensure we have lots of entries:
sage: len(sloane.trait_names()) > 100 True
And ensure none are being incorrectly returned:
sage: [ None for n in sloane.trait_names() if not n.startswith('A') ] []
Ensure we can access dynamic constructions and cache correctly:
sage: s = sloane.A000587 sage: s is sloane.A000587 True
And that we can access other functions in parent classes:
sage: sloane.__class__ <class 'sage.combinat.sloane_functions.Sloane'>
Author: Nick Alexander
Functions: trait_names
self) |
The member classes are inspected from module sage.combinat.sloane_functions.
They must be sub classes of SloaneSequence and must start with 'A'. These restrictions are only to prevent typos, incorrect inspecting, etc.
sage: type(sloane.trait_names()) <type 'list'>
Special Functions: __getattribute__
self, name) |
sage: sloane.__getattribute__('A000001') Number of groups of order n. sage: sloane.__getattribute__('dog') Traceback (most recent call last): ... AttributeError: dog
Class: SloaneSequence
We create a dummy sequence:
self, [offset=1]) |
A sequence starting at offset (=1 by default).
sage: from sage.combinat.sloane_functions import SloaneSequence sage: SloaneSequence().offset 1 sage: SloaneSequence(4).offset 4
Functions: list
self, n) |
sage: sloane.A000012.list(4) [1, 1, 1, 1]
Special Functions: __call__,
__cmp__,
__getitem__,
__init__,
__iter__,
_eval,
_repr_
self, n) |
sage: sloane.A000007(2) 0 sage: sloane.A000007('a') Traceback (most recent call last): ... TypeError: input must be an int, long, or Integer sage: sloane.A000007(-1) Traceback (most recent call last): ... ValueError: input n (=-1) must be an integer >= 0 sage: sloane.A000001(0) Traceback (most recent call last): ... ValueError: input n (=0) must be a positive integer
self, other) |
sage: cmp(sloane.A000007,sloane.A000045) == 0 False sage: cmp(sloane.A000007,sloane.A000007) == 0 True
self, n) |
We interpret slices as best we can, but our sequences are infinite so we want to prevent some mis-incantations.
Therefore, we abitrarily cap slices to be at most LENGTH=100000 elements long. Since many Sloane sequences are costly to compute, this is probably not an unreasonable decision, but just in case, list does not cap length.
sage: sloane.A000012[3] 1 sage: sloane.A000012[:4] [1, 1, 1, 1] sage: sloane.A000012[:10] [1, 1, 1, 1, 1, 1, 1, 1, 1, 1] sage: sloane.A000012[4:10] [1, 1, 1, 1, 1, 1] sage: sloane.A000012[0:1000000000] Traceback (most recent call last): ... IndexError: slice (=slice(0, 1000000000, None)) too long
self) |
sage: iter(sloane.A000012) Traceback (most recent call last): ... NotImplementedError
self, n) |
sage: from sage.combinat.sloane_functions import SloaneSequence sage: SloaneSequence(0)._eval(4) Traceback (most recent call last): ... NotImplementedError
self) |
sage: from sage.combinat.sloane_functions import SloaneSequence sage: SloaneSequence(4)._repr_() Traceback (most recent call last): ... NotImplementedError
See About this document... for information on suggesting changes.