

A000012


The simplest sequence of positive numbers: the all 1's sequence.
(Formerly M0003)


2229



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
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OFFSET

0,1


COMMENTS

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 Deléham, Jul 07 2005
A063524(a(n)) = 1.  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)).  K.V.Iyer, Apr 11 2009
The partial sums give the natural numbers (A000027).  Daniel Forgues, May 08 2009
From Enrique Pérez Herrero, 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 squarefree and a perfect square. See A005117 and A000290. (End)
Also smallest divisor of n.  JuriStepan Gerasimov, Sep 07 2009
Also decimal expansion of 1/9.  Enrique Pérez Herrero, Sep 18 2009; corrected by Klaus Brockhaus, Apr 02 2010
a(n) is also the number of complete graphs on n nodes.  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(p1) for prime p.  Jaroslav Krizek, Oct 18 2009
nth prime minus phi(prime(n)); number of divisors of nth prime minus number of perfect partitions of nth prime; the number of perfect partitions of nth prime number; the number of perfect partitions of nth noncomposite number.  JuriStepan Gerasimov, Oct 26 2009
For all n>0, the sequence of limit values for a(n) = n!*Sum_{k>=n} k/(k+1)!. Also, a(n) = n^0.  Harlan J. Brothers, Nov 01 2009
a(n) is also the number of 0regular graphs on n vertices.  Jason Kimberley, Nov 07 2009
Differences between consecutive n.  JuriStepan Gerasimov, Dec 05 2009
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 kth power in the expansion of (x^(n+1)1)/(x1).
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 qnomial array, T(n,k) is the coefficient of the kth power in the expansion of ((x^q 1)/(x1))^n, and row n has a sum of q^n and a length of (q1)*n + 1. (End)
The number of maximal selfavoiding walks from the NW to SW corners of a 2 X 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.)  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.  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 nth diagonal by t^n gives M(t) = I/(It*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 subdiagonal of T by t and the other subdiagonals 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
The original definition of the meter was one tenmillionth of the distance from the Earth's equator to the North Pole. According to that historical definition, the length of one degree of latitude, that is, 60 nautical miles, would be exactly 111111.111... meters.  JeanFrançois Alcover, Jun 02 2013
Deficiency of 2^n.  Omar E. Pol, Jan 30 2014
Consider n >= 1 nonintersecting spheres each with surface area S. Define point p on sphere S_i to be a "public point" if and only if there exists a point q on sphere S_j, j != i, such that line segment pq INTERSECT S_i = {p} and pq INTERSECT S_j = {q}; otherwise, p is a "private point". The total surface area composed of exactly all private points on all n spheres is a(n)*S = S. ("The Private Planets Problem" in Zeitz.)  Rick L. Shepherd, May 29 2014
For n>0, digital roots of centered 9gonal numbers (A060544).  Colin Barker, Jan 30 2015
Product of nonzero digits in base 2 representation of n.  Franklin T. AdamsWatters, May 16 2016
Alternating row sums of triangle A104684.  Wolfdieter Lang, Sep 11 2016
A fixed point of the run length transform.  Chai Wah Wu, Oct 21 2016
Length of period of continued fraction for sqrt(A002522) or sqrt(A002496).  A.H.M. Smeets, Oct 10 2017
a(n) is also the determinant of the (n+1) X (n+1) matrix M defined by M(i,j) = binomial(i,j) for 0 <= i,j <= n, since M is a lower triangular matrix with main diagonal all 1's.  Jianing Song, Jul 17 2018
a(n) is also the determinant of the symmetric n X n matrix M defined by M(i,j) = min(i,j) for 1 <= i,j <= n (see Xavier Merlin reference).  Bernard Schott, Dec 05 2018
a(n) is also the determinant of the symmetric n X n matrix M defined by M(i,j) = tau(gcd(i,j)) for 1 <= i,j <= n (see De Koninck & Mercier reference).  Bernard Schott, Dec 08 2020


REFERENCES

J.M. De Koninck & A. Mercier, 1001 Problèmes en Théorie Classique des Nombres, Problème 692 pp. 90 and 297, Ellipses, Paris, 2004.
L. B. W. Jolley, Summation of Series, Second Revised Edition, Dover (1961).
Xavier Merlin, Méthodix Algèbre, Exercice 1a), page 153, Ellipses, Paris, 1995.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 55.
Paul Zeitz, The Art and Craft of Mathematical Problem Solving, The Great Courses, The Teaching Company, 2010 (DVDs and Course Guidebook, Lecture 6: "Pictures, Recasting, and Points of View", pp. 3234).


LINKS

Charles R Greathouse IV, Table of n, a(n) for n = 0..10000 [Useful when plotting one sequence against another. See Swayne link.]
Jeremiah Bartz, Bruce Dearden, and Joel Iiams, Classes of Gap Balancing Numbers, arXiv:1810.07895 [math.NT], 2018.
Harlan Brothers, Factorial: Summation (formula 06.01.23.0002), The Wolfram Functions Site  Harlan J. Brothers, Nov 01 2009
Daniele A. Gewurz and Francesca Merola, Sequences realized as Parker vectors of oligomorphic permutation groups, J. Integer Seqs., Vol. 6, 2003.
A. M. Hinz, S. Klavžar, U. Milutinović, and C. Petr, The Tower of Hanoi  Myths and Maths, Birkhäuser 2013. See page 172. Book's website
Jerry Metzger and Thomas Richards, A Prisoner Problem Variation, Journal of Integer Sequences, Vol. 18 (2015), Article 15.2.7.
László Németh, The trinomial transform triangle, J. Int. Seqs., Vol. 21 (2018), Article 18.7.3. Also arXiv:1807.07109 [math.NT], 2018.
Robert Price, Comments on A000012 concerning Elementary Cellular Automata, Jan 31 2016
N. J. A. Sloane, Illustration of initial terms
Michael Z. Spivey and Laura L. Steil, The kBinomial Transforms and the Hankel Transform, Journal of Integer Sequences, Vol. 9 (2006), Article 06.1.1.
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
Eric Weisstein's World of Mathematics, Elementary Cellular Automaton
S. Wolfram, A New Kind of Science
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 partitioncounting sequences
Index entries for linear recurrences with constant coefficients, signature (1).
Index entries for sequences that are fixed points of mappings


FORMULA

a(n) = 1.
G.f.: 1/(1x).
E.g.f.: exp(x).
G.f.: Product_{k>=0} (1 + x^(2^k)).  Zak Seidov, Apr 06 2007
Completely multiplicative with a(p^e) = 1.
Regarded as a square array by antidiagonals, g.f. 1/((1x)(1y)), 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/(1x). Regarded as a triangular array, g.f. 1/((1x)(1xy)), e.g.f. Sum T(n,m) x^n y^m/m! = e^{xy}/(1x).  Franklin T. AdamsWatters, Feb 06 2006
Dirichlet g.f.: zeta(s).  Ilya Gutkovskiy, Aug 31 2016
a(n) = Sum_{l=1..n} (1)^(l+1)*2*cos(Pi*l/(2*n+1)) = 1 identically in n >= 1 (for n=0 one has 0 from the undefined sum). From the Jolley reference, (429) p. 80. Interpretation: consider the n segments between x=0 and the n positive zeros of the Chebyshev polynomials S(2*n, x) (see A049310). Then the sum of the lengths of every other segment starting with the one ending in the largest zero (going from the right to the left) is 1.  Wolfdieter Lang, Sep 01 2016
As a lower triangular matrix, T = M*T^(1)*M = M*A167374*M, where M(n,k) = (1)^n A130595(n,k). Note that M = M^(1). Cf. A118800 and A097805.  Tom Copeland, Nov 15 2016


EXAMPLE

1 + 1/(1 + 1/(1 + 1/(1 + 1/(1 + ...)))) = A001622.
1/9 = 0.11111111111111...
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)


MAPLE

seq(1, i=0..150);


MATHEMATICA

Array[1 &, 50] (* Joseph Biberstine (jrbibers(AT)indiana.edu), Dec 26 2006 *)


PROG

(Magma) [1 : n in [0..100]];
(PARI) {a(n) = 1};
(Haskell)
a000012 = const 1
a000012_list = repeat 1  Reinhard Zumkeller, May 07 2012
(Maxima) makelist(1, n, 1, 30); /* Martin Ettl, Nov 07 2012 */
(Python) print([1 for n in range(90)]) # Michael S. Branicky, Apr 04 2022


CROSSREFS

Cf. A000004, A007395, A010701, A000027, A027641, A014410, A211216, A212393, A060544, A051801, A104684.
For other qnomial arrays, see A007318, A027907, A008287, A035343, A063260, A063265, A171890.  Matthew Vandermast, Oct 31 2010
Cf. A097805, A118800, A130595, A167374, A008284 (multisets).
Sequence in context: A155040 A209661 A033999 * A216430 A232544 A309873
Adjacent sequences: A000009 A000010 A000011 * A000013 A000014 A000015


KEYWORD

nonn,core,easy,mult,cofr,cons,tabl,changed


AUTHOR

N. J. A. Sloane, May 16 1994


STATUS

approved



