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A000012
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The simplest sequence of positive numbers: the all 1's sequence.
(Formerly M0003)
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1186
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1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1
(list;
table;
constant;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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0,1
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COMMENTS
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Number of ways of writing n as a product of primes.
Number of ways of writing n as a sum of distinct powers of 2.
Continued fraction for golden ratio A001622.
Partial sums of A000007 (characteristic function of 0). - Jeremy Gardiner, Sep 08 2002
An example of an infinite sequence of positive integers whose distinct pairwise concatenations are all primes! - Don Reble, Apr 17 2005
Binomial transform of A000007; inverse binomial transform of A000079 . Philippe DELEHAM, Jul 07 2005
A063524(a(n)) = 1. [From Reinhard Zumkeller, Oct 11 2008]
For n >= 0, let M(n) be the matrix with first row = (n n+1) and 2nd row = (n+1 n+2). Then a(n) = absolute value of det(M(n)). [From Kailasam Viswanathan Iyer (kvi(AT)nitt.edu), Apr 11 2009]
The partial sums give the natural numbers (A000027). [From Daniel Forgues, May 08 2009]
Contribution from Barbarel Tres Mil (barbarel3000(AT)yahoo.es), Sep 04 2009: (Start)
a(n) is also tau_1(n) where tau_2(n) is A000005
a(n) is a completely multiplicative arithmetical function.
a(n) is both square free and a perfect square. See A005117 and A000290. (End)
Also smallest divisor of n. [From Juri-Stepan Gerasimov, Sep 07 2009].
Also decimal expansion of 1/9. [From Barbarel Tres Mil (barbarel3000(AT)yahoo.es), Sep 18 2009; corrected by Klaus Brockhaus, Apr 02 2010]
a(n) is also the number of complete graphs on n nodes. [From Pablo Chavez (pchavez(AT)cmu.edu), Sep 15 2009]
Totally multiplicative sequence with a(p) = 1 for prime p. Totally multiplicative sequence with a(p) = a(p-1) for prime p. [From Jaroslav Krizek, Oct 18 2009]
n-th prime minus phi(prime(n)); number of divisors of n-th prime minus number of perfect partitions of n-th prime; the number of perfect partitions of n-th prime number; the number of perfect partitions of n-th non-composite number. [From Juri-Stepan Gerasimov, Oct 26 2009]
Contribution from Harlan J. Brothers, Nov 01 2009: (Start)
For all n>0, the sequence of limit values for a(n)=n!Sum[k=n..inf, k/(k+1)! ]
Also, for all n != 0, a(n)=n^0 (End)
a(n) is also the number of 0-regular graphs on n vertices. [From Jason Kimberley, Nov 07 2009]
Differences between consecutive n. [From Juri-Stepan Gerasimov, Dec 05 2009]
Contribution from Matthew Vandermast, Oct 31 2010: (Start)
1) When sequence is read as a regular triangular array, T(n,k) is the coefficient of the k-th power in the expansion of (x^(n+1)-1)/(x-1).
2) Sequence can also be read as a uninomial array with rows of length 1, analogous to arrays of binomial, trinomial, etc., coefficients. In a q-nomial array, T(n,k) is the coefficient of the k-th power in the expansion of ((x^q -1)/(x-1))^n, and row n has a sum of q^n and a length of (q-1)*n + 1. (End)
The number of maximal self-avoiding walks from the NW to SW corners of a 2-by-n grid.
When considered as a rectangular array, A000012 is a member of the chain of accumulation arrays that includes the multiplication table A003991 of the positive integers. The chain is ... < A185906 < A000007 < A000012 < A003991 < A098358 < A185904 < A185905 < ... (See A144112 for the definition of accumulation array.) [From Clark Kimberling, Feb 06 2011]
a(n) = A007310(n+1) (Modd 3) := A193680(A007310(n+1)), n>=0. For general Modd n (not to be confused with mod n) see a comment on A203571. The nonnegative members of the three residue classes Modd 3, called [0], [1], and [2], are shown in the array A088520, if there the third row is taken as class [0] after inclusion of 0. [From Wolfdieter Lang, Feb 09 2012]
Let M = Pascal's triangle without 1's (A014410) and V = a variant of the Bernoulli numbers A027641 but starting [1/2, 1/6, 0, -1/30,...]. Then M*V = [1, 1, 1, 1,...]. - Gary W. Adamson, Mar 05 2012
As a lower triangular array, T is an example of the fundamental generalized factorial matrices of A133314. Multiplying each n-th diagonal by t^n gives M(t) = I/(I-t*S) = I + t*S + (t*S)^2 + ... where S is the shift operator A129184, and T = M(1). The inverse of M(t) is obtained by multiplying the first sub-diagonal of T by -t and the other sub-diagonals by zero, so A167374 is the inverse of T. Multiplying by t^n/n! gives exp(t*S) with inverse exp(-t*S). - Tom Copeland, Nov 10 2012
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REFERENCES
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N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
Michael Z. Spivey and Laura L. Steil, The k-Binomial Transforms and the Hankel Transform, Journal of Integer Sequences, Vol. 9 (2006), Article 06.1.1.
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LINKS
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N. J. A. Sloane, Table of n, a(n) for n = 0..1000 [Useful when plotting one sequence against another. See Swayne link.]
David Applegate, Omar E. Pol and N. J. A. Sloane, The Toothpick Sequence and Other Sequences from Cellular Automata
Harlan Brothers, Factorial: Summation (formula 06.01.23.0002), The Wolfram Functions Site [From Harlan J. Brothers, Nov 01 2009]
Daniele A. Gewurz and Francesca Merola, Sequences realized as Parker vectors ..., J. Integer Seqs., Vol. 6, 2003.
A. M. Hinz, S. Klavžar, U. Milutinović, C. Petr, The Tower of Hanoi - Myths and Maths, Birkhäuser 2013. See page 172. Book's website
N. J. A. Sloane, Illustration of initial terms
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
D. F. Swayne, Plot pairs of sequences in the OEIS
Eric Weisstein's World of Mathematics, Golden Ratio
Eric Weisstein's World of Mathematics, Chromatic Number
Eric Weisstein's World of Mathematics, Graph Cycle
G. Xiao, Contfrac
Index entries for "core" sequences
Index entries for characteristic functions
Index entries for continued fractions for constants
Index to divisibility sequences
Index entries for related partition-counting sequences
Index entries for sequences related to linear recurrences with constant coefficients
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FORMULA
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G.f.: 1/(1-x); a(n)=1. E.g.f.: e^x.
G.f.: Product[(1+x^(2^k)),{k,0,Infinity}]. - Zak Seidov, Apr 06 2007
Completely multiplicative with a(p^e) = 1.
Dirichlet generating function: zeta(s). - Franklin T. Adams-Watters, Sep 11 2005.
Regarded as a square array by antidiagonals, g.f. 1/((1-x)(1-y)), e.g.f. sum T(n,m) x^n/n! y^m/m! = e^{x+y}, e.g.f. sum T(n,m) x^n y^m/m! = e^y/(1-x). Regarded as a triangular array, g.f. 1/((1-x)(1-xy)), e.g.f. sum T(n,m) x^n y^m/m! = e^{xy}/(1-x). - Franklin T. Adams-Watters, Feb 06 2006
Contribution from Barbarel Tres Mil (barbarel3000(AT)yahoo.es), Sep 04 2009: (Start)
a(n) = Sum(d|n,mu(n/d)*tau_2(d))=1, where tau_2(n)=A000005 and mu(n)=A008683
a(n) = |Sum(d|n,mu(d)*tau_2(d))|=1 (End)
a(n) = A002033(A000040(n)) = A002033(A008578(n)) = A000005(A000040(n)) -A002033(n) = A000027(A000040(n)) -A000010(A000040(n)). [From Juri-Stepan Gerasimov, Oct 26 2009]
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EXAMPLE
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1.618033988749894848204586834... = 1 + 1/(1 + 1/(1 + 1/(1 + 1/(1 + ...))))
From Wolfdieter Lang, Feb 09 2012, (Start)
Modd 7 for nonnegative odd numbers not divisible by 3:
A007310: 1, 5, 7, 11, 13, 17, 19, 23, 25, 29, 31, 35, 37, ...
Modd 3: 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
(End)
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MAPLE
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A000012 := n->1;
[ seq(1, i=0..100) ];
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MATHEMATICA
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a[n_] := 1
Array[1 &, 50] - Joseph Biberstine (jrbibers(AT)indiana.edu), Dec 26 2006
Table[n!Sum[k/(k+1)!, {k, n, \[Infinity]}], {n, 10}] [From Harlan J. Brothers, Nov 01 2009]
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PROG
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(MAGMA) [1 : n in [0..100]];
(PARI) a(n)=1
(PARI) { default(realprecision, 1080); phi = (1 + sqrt(5))/2; x=contfrac(phi); for (n=1, 1001, write("b000012.txt", n-1, " ", x[n])); } [From Harry J. Smith, May 14 2009]
(Haskell)
a000012 = const 1
a000012_list = repeat 1 -- Reinhard Zumkeller, May 07 2012
(Maxima) makelist(1, n, 1, 30); /* Martin Ettl, Nov 07 2012 */
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CROSSREFS
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Cf. A000004, A007395, A010701, A000027.
Cf. tau_2(n): A000005, tau_3(n): A007425, tau_4(n): A007426, tau_5(n): A061200, tau_6(n): A034695, (unordered) 2-factorizations of n: A038548, (unordered) 3-factorizations of n: A034836, A001055, (tau<=)_2(n): A006218, (tau<=)_3(n): A061201, (tau<=)_5(n): A061203, (tau<=)_6(n): A061204. [From Barbarel Tres Mil (barbarel3000(AT)yahoo.es), Sep 04 2009]
Regular graphs A005176 (any degree), A051031 (triangular array), chosen degrees: A000012 (k=0), A059841 (k=1), A008483 (k=2), A005638 (k=3), A033301 (k=4), A165626 (k=5), A165627 (k=6), A165628 (k=7). [From Jason Kimberley, Nov 07 2009]
For other q-nomial arrays, see A007318, A027907, A008287, A035343, A063260, A063265, A171890. [From Matthew Vandermast, Oct 31 2010]
Cf. A027641, A014410.
Cf. A211216, A212393.
Sequence in context: A076479 A155040 A033999 * A162511 A157895 A063747
Adjacent sequences: A000009 A000010 A000011 * A000013 A000014 A000015
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KEYWORD
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nonn,core,easy,mult,cofr,cons,tabl
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AUTHOR
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N. J. A. Sloane.
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STATUS
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approved
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