A267142 The characteristic function of the multiples of 9.

1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0

Offset

0

Comments

Decimal expansion of 100000000/999999999.

Period 9: repeat [1, 0, 0, 0, 0, 0, 0, 0, 0].

More generally, the ordinary generating function for the characteristic function of the multiples of k is 1/(1 - x^k).

Links

Antti Karttunen, Table of n, a(n) for n = 0..999

Index entries for characteristic functions

Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,0,0,1).

Formula

G.f.: 1/(1 - x^9) = -1 / ( (x-1)*(1+x+x^2)*(x^6+x^3+1) ).

a(n) = abs(sign(n mod 9) - 1).

a(n) = abs(A168182(n)-1).

A007953(a(n)*n) mod 9 = 0.

Mathematica

Table[Boole[Divisible[n, 9]], {n, 0, 115}]
Table[Abs[Sign[Mod[n, 9]] - 1], {n, 0, 115}]
CoefficientList[Series[1 / (1 - x^9), {x, 0, 100}], x] (* Vincenzo Librandi, Jan 11 2016 *)

Prog

(Magma) &cat[&cat[[1], [0]^^8]^^14]; // Vincenzo Librandi, Jan 11 2016
(PARI) a(n) = n\9 - (n-1)\9; \\ Altug Alkan, Jan 11 2016
(PARI) A267142(n) = !(n%9); \\ Antti Karttunen, Oct 07 2017

Crossrefs

Cf. A008591, A079978, A079979, A079998, A082784, A121262, A168182, A253513.

Sequence in context: A014954 A015899 A015494 * A185119 A280130 A304002

Adjacent sequences: A267139 A267140 A267141 * A267143 A267144 A267145

Keyword

nonn,easy

Author

Ilya Gutkovskiy, Jan 11 2016

Status

approved