

A000012


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


2352



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.
An example of an infinite sequence of positive integers whose distinct pairwise concatenations are all primes!  Don Reble, Apr 17 2005
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
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)
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
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.
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
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
A fixed point of the run length transform.  Chai Wah Wu, Oct 21 2016
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



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
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


EXAMPLE

1 + 1/(1 + 1/(1 + 1/(1 + 1/(1 + ...)))) = A001622.
1/9 = 0.11111111111111...
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
(Maxima) makelist(1, n, 1, 30); /* Martin Ettl, Nov 07 2012 */


CROSSREFS

Cf. A000004, A007395, A010701, A000027, A027641, A014410, A211216, A212393, A060544, A051801, A104684.


KEYWORD



AUTHOR



STATUS

approved



