A097325 Period 6: repeat [0, 1, 1, 1, 1, 1].

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

Offset: 0

Author

Ralf Stephan, Aug 16 2004

Keywords

• nonn
• easy

Comments

a(n) is 0 if 6 divides n, 1 otherwise.

Links

• Antti Karttunen, Table of n, a(n) for n = 0..26244
• Index entries for characteristic functions
• Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,1).

Crossrefs

Characteristic sequence of A047253.

Binary complement of A079979.

Cf. A010875, A168185, A145568, A168184, A168182, A168181, A109720, A011558, A166486, A011655, A000035.

Programs

Magma

[Sign(n mod 6) : n in [0..50]]; // Wesley Ivan Hurt, Jun 20 2014

Maple

seq(signum(k mod 6), k=0..100); # Wesley Ivan Hurt, Jun 29 2013

Mathematica

Table[Boole[Not[Divisible[n, 6]]], {n, 0, 89}] (* Alonso del Arte, Oct 21 2013 *)
PadRight[{}, 120, {0, 1, 1, 1, 1, 1}] (* Michael De Vlieger, Dec 22 2017 *)

PARI

a(n) = sign(n%6);

Scheme

(define (A097325 n) (if (zero? (modulo n 6)) 0 1)) ;; Antti Karttunen, Dec 22 2017

Formula

G.f.: 1/(1-x) - 1/(1-x^6) = Sum_{k>=0} x^k - x^(6*k).

Recurrence: a(n+6) = a(n), a(0) = 0, a(i) = 1, 1 <= i <= 5.

a(n) = (1/4) * (3 - (-1)^n - (-1)^((n+1)/3) - (-1)^((2n+1)/3)).

From Reinhard Zumkeller, Nov 30 2009: (Start)

a(n) = 1 - A079979(n).

a(A047253(n)) = 1, a(A008588(n)) = 0.

A033438(n) = Sum_{k=0..n} a(k)*(n-k). (End)

Dirichlet g.f.: (1 - 1/6^s)*zeta(s). - R. J. Mathar, Feb 19 2011

For the general case: the characteristic function of numbers that are not multiples of m is a(n) = floor((n-1)/m) - floor(n/m) + 1, m, n > 0. - Boris Putievskiy, May 08 2013

a(n) = sign(n mod 6). - Wesley Ivan Hurt, Jun 29 2013

a(n) = ceiling(5n/6) - floor(5n/6). - Wesley Ivan Hurt, Jun 20 2014

Extensions

New name from Omar E. Pol, Oct 21 2013