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

Author

Paul Barry, Feb 02 2005

Keywords

• easy
• sign

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

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

Crossrefs

Cf. A131719.

Programs

Magma

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

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