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A000007 The characteristic function of 0: a(n) = 0^n.
(Formerly M0002)
513
1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 (list; constant; graph; refs; listen; history; text; internal format)
OFFSET

0,1

COMMENTS

Changing the offset to 1 gives the arithmetical function a(1) = 1, a(n) = 0 for n > 1, the identity function for Dirichlet multiplication (see Apostol). - N. J. A. Sloane

Changing the offset to 1 makes this the decimal expansion of 1. - N. J. A. Sloane, Nov 13 2014

Hankel transform (see A001906 for definition) of : A000007 (powers of 0), A000012 (powers of 1), A000079 (powers of 2), A000244 (powers of 3), A000302 (powers of 4), A000351 (powers of 5), A000400 (powers of 6), A000420 (powers of 7), A001018 (powers of 8), A001019 (powers of 9), A011557 (powers of 10), A001020 (powers of 11), etc. ... - Philippe Deléham, Jul 07 2005

This is the identity sequence with respect to convolution. - David W. Wilson, Oct 30 2006

a(A000004(n)) = 1; a(A000027(n)) = 0. - Reinhard Zumkeller, Oct 12 2008

The alternating sum of the n-th row of Pascal's triangle gives the characteristic function of 0, a(n) = 0^n. - Daniel Forgues, May 25 2010

The number of maximal self-avoiding walks from the NW to SW corners of a 1 X n grid. - Sean A. Irvine, Nov 19 2010

Historically there has been some disagreement as to whether 0^0 = 1. Graphing x^0 seems to support that conclusion, but graphing 0^x instead suggests that 0^0 = 0. Euler and Knuth have argued in favor of 0^0 = 1. For some calculators, 0^0 triggers an error, while in Mathematica, 0^0 is Indeterminate. - Alonso del Arte, Nov 15 2011

Another consequence of changing the offset to 1 is that then this sequence can be described as the sum of Moebius mu(d) for the divisors d of n. - Alonso del Arte, Nov 28 2011

With the convention 0^0 = 1, 0^n = 0 for n > 0, the sequence a(n) = 0^|n-k|, which equals 1 when n = k and is 0 for n >= 0, has g.f. x^k. A000007 is the case k = 0. - George F. Johnson, Mar 08 2013

A fixed point of the run length transform. - Chai Wah Wu, Oct 21 2016

REFERENCES

T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 30.

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.

LINKS

David Wasserman, Table of n, a(n) for n = 0..1000

Paul Barry, A Catalan Transform and Related Transformations on Integer Sequences, Journal of Integer Sequences, Vol. 8 (2005), Article 05.4.5.

Paul Barry, A Note on a Family of Generalized Pascal Matrices Defined by Riordan Arrays, Journal of Integer Sequences, 16 (2013), #13.5.4.

Dr. Math, 0^0 (zero to the zero power)

Daniele A. Gewurz and Francesca Merola, Sequences realized as Parker vectors of oligomorphic permutation groups, J. Integer Seqs., Vol. 6, 2003.

Donald E. Knuth, Two notes on notation, arXiv:math/9205211 [math.HO], 1992. See page 6 on 0^0.

Robert Price, Comments on A000007, Jan 27 2016

Eric Weisstein's World of Mathematics, Elementary Cellular Automaton

S. Wolfram, A New Kind of Science

Index entries for "core" sequences

Index entries for characteristic functions

Index entries for linear recurrences with constant coefficients signature (1).

Index entries for sequences related to cellular automata

Index to Elementary Cellular Automata

FORMULA

Multiplicative with a(p^e) = 0. - David W. Wilson, Sep 01 2001

a(n) = floor(1/(n + 1)). - Franz Vrabec, Aug 24 2005

a(n) = ((n + 1)!^2 mod (n + 2))*((n + 2)!^2 mod (n + 3)), with n >= 0. - Paolo P. Lava, Apr 24 2007

a(n) = 1 - (((n + 1)! + 1) mod (n + 1)). - Paolo P. Lava, May 22 2007

a(n) = 1 - ((n + 2) mod (n + 1)). - Paolo P. Lava, Jun 27 2007

a(n) = C(2*n, n) mod 2. - Paolo P. Lava, Aug 31 2007

As a function of Bernoulli numbers, (Cf. A027641: (1, -1/2, 1/6, 0, -1/30, ...)); triangle A074909 (the beheaded Pascal's triangle) * B_n as a vector = [1, 0, 0, 0, 0,...]. - Gary W. Adamson, Mar 05 2012

a(n) = Sum_{k = 0..n} exp(2*Pi*i*k/(n+1)) is the sum of the (n+1)th roots of unity. - Franz Vrabec, Nov 09 2012

a(n) = (1-(-1)^(2^n))/2. - Luce ETIENNE, May 05 2015

a(n) = 1 - A057427(n). - Alois P. Heinz, Jan 20 2016

From Ilya Gutkovskiy, Sep 02 2016: (Start)

Binomial transform of A033999.

Inverse binomial transform of A000012. (End)

EXAMPLE

a(4) = 0 = (1, 5, 10, 10, 5) dot (1, -1/2, 1/6 0, -1/30) = (1 - 5/2 + 5/3 + 0 - 1/6) = 0; where (1, 5, 10, 10, 5) = row 4 of triangle A074909. - Gary W. Adamson, Mar 05 2012

MAPLE

A000007 := proc(n) if n = 0 then 1 else 0; fi; end;

with(combstruct); spec := [A, {A=Z} ]; [seq(combstruct[count](spec, size=n), n=1..20)];

MATHEMATICA

Table[If[n == 0, 1, 0], {n, 0, 99}]

Table[Boole[n == 0], {n, 0, 99}] (* Michael Somos, Aug 25 2012 *)

Join[{1}, LinearRecurrence[{1}, {0}, 102]] (* Ray Chandler, Jul 30 2015 *)

PROG

(PARI) {a(n) = !n};

(MAGMA) [1] cat [0:n in [1..100]]; // Sergei Haller, Dec 21 2006

(Haskell)

a000007 = (0 ^)

a000007_list = 1 : repeat 0

-- Reinhard Zumkeller, May 07 2012, Mar 27 2012

CROSSREFS

Characteristic function of g: this sequence (g = 0), A063524 (g = 1), A185012 (g = 2), A185013 (g = 3), A185014 (g = 4), A185015 (g = 5), A185016 (g = 6), A185017 (g = 7). - Jason Kimberley, Oct 14 2011

Characteristic function of multiples of g: this sequence (g = 0), A000012 (g = 1), A059841 (g = 2), A079978 (g = 3), A121262 (g = 4), A079998 (g = 5), A079979 (g = 6), A082784 (g = 7). - Jason Kimberley, Oct 14 2011

Cf. A074909, A027641, A057427.

Sequence in context: A185013 A185012 A185017 * A014041 A015868 A015824

Adjacent sequences:  A000004 A000005 A000006 * A000008 A000009 A000010

KEYWORD

core,nonn,mult,cons,easy

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified December 6 15:11 EST 2016. Contains 278781 sequences.