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A000012 The simplest sequence of positive numbers: the all 1's sequence.
(Formerly M0003)
1280
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)
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

Contribution 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 square free and a perfect square. See A005117 and A000290. (End)

Also smallest divisor of n. - Juri-Stepan 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(p-1) for prime p. - 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. - Juri-Stepan Gerasimov, Oct 26 2009

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. - Harlan J. Brothers, Nov 01 2009

a(n) is also the number of 0-regular graphs on n vertices. - Jason Kimberley, Nov 07 2009

Differences between consecutive n. - Juri-Stepan 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 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 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 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

The original definition of the meter was one ten-millionth 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. - Jean-François Alcover, Jun 02 2013

Deficiency of 2^n. - Omar E. Pol, Jan 30 2014

REFERENCES

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

LINKS

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 of oligomorphic permutation groups, 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

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.

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 linear recurrences with constant coefficients, signature (1).

FORMULA

a(n)=1.

G.f.: 1/(1-x).

E.g.f.: exp(x).

G.f.: prod( k>=0, 1 + x^(2^k) ). - 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

E.g.f.: E(0)-1-x, where E(k) = 2 + 2*x - 8*k^2 + x^2*(2*k+3)*(2*k-1)/E(k+1) ; (continued fraction). - Sergei N. Gladkovskii, Dec 20 2013

E.g.f.: E(0)-1, where E(k) = 2 + x/(2*k+1 - x/E(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Dec 23 2013

EXAMPLE

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)

MAPLE

A000012 := n->1;

[ seq(1, i=0..100) ];

MATHEMATICA

a[n_] := 1

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

PROG

(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])); } \\ 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 */

CROSSREFS

Cf. A000004, A007395, A010701, A000027.

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). - Jason Kimberley, Nov 07 2009

For other q-nomial arrays, see A007318, A027907, A008287, A035343, A063260, A063265, A171890. - 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

KEYWORD

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

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified April 18 10:47 EDT 2014. Contains 240707 sequences.