A103368 Period 6: repeat [1, 1, -1, -1, 0, 0].

1, 1, -1, -1, 0, 0, 1, 1, -1, -1, 0, 0, 1, 1, -1, -1, 0, 0, 1, 1, -1, -1, 0, 0, 1, 1, -1, -1, 0, 0, 1, 1, -1, -1, 0, 0, 1, 1, -1, -1, 0, 0, 1, 1, -1, -1, 0, 0, 1, 1, -1, -1, 0, 0, 1, 1, -1, -1, 0, 0, 1, 1, -1, -1, 0, 0, 1, 1, -1, -1, 0, 0, 1, 1, -1, -1, 0, 0, 1, 1, -1, -1, 0, 0

Offset

0,1

Comments

The positive sequence is A131719(n+1) = a(n) = (cos(2*Pi*n/3+Pi/3)/6+sqrt(3)*sin(2*Pi*n/3+Pi/3)/6 -sqrt(3)*cos(Pi*n/3+Pi/6)/6+sin(Pi*n/3+Pi/6)/2+2/3, with g.f. (1+x^2) / ( (1-x)*(1-x+x^2)*(1+x+x^2) ).

Links

Table of n, a(n) for n=0..83.

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

Formula

G.f.: (1+x)/(1+x^2+x^4).

a(n) = Sum_{k=0..floor(n/2)} binomial(k, floor(n/2)-k)*(-1)^k.

a(n) = -cos(2*Pi*n/3+Pi/3)/2 + sqrt(3)*sin(2*Pi*n/3+Pi/3)/6 + sqrt(3)*cos(Pi*n/3+Pi/6)/2 + sin(Pi*n/3+Pi/6)/2.

a(n) = cos(Pi*n/3) + sin(2*Pi*n/3)/sqrt(3). - R. J. Mathar, Oct 08 2011

a(n) + a(n-2) + a(n-4) = 0 for n>3. - Wesley Ivan Hurt, Jun 20 2016

E.g.f.: (sqrt(3)*sin(sqrt(3)*x/2) + 3*cos(sqrt(3)*x/2)*exp(x))*exp(-x/2)/3. - Ilya Gutkovskiy, Jun 21 2016

Maple

A103368:=n->[1, 1, -1, -1, 0, 0][(n mod 6)+1]: seq(A103368(n), n=0..100); # Wesley Ivan Hurt, Jun 20 2016

Mathematica

PadRight[{}, 100, {1, 1, -1, -1, 0, 0}] (* Wesley Ivan Hurt, Jun 20 2016 *)

Prog

(Magma) &cat [[1, 1, -1, -1, 0, 0]^^20]; // Wesley Ivan Hurt, Jun 20 2016

Crossrefs

Cf. A131719.

Sequence in context: A130093 A115944 A166446 * A055132 A285403 A183918

Adjacent sequences: A103365 A103366 A103367 * A103369 A103370 A103371

Keyword

easy,sign

Author

Paul Barry, Feb 02 2005

Status

approved