A163804 Expansion of (1 - x) * (1 - x^4) / ((1 - x^2) * (1 - x^3)) in powers of x.

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

Offset

0,1

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000

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

Formula

Euler transform of length 4 sequence [ -1, 1, 1, -1].

G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = 2 - v - u * (4 - 2*v - u).

a(3*n) = 0 unless n=0, a(3*n + 1) = -1, a(3*n + 2) = a(0) = 1.

a(-n) = -a(n) unless n=0. a(n+3) = a(n) unless n=0 or n=-3.

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

G.f.: A(x) = 1 / (1 + x / (1 + x^2)) = 1 - x / (1 + x / (1 - x / (1 + x))). - Michael Somos, Jan 03 2013

a(n) = A057078(n-2), n>1. - R. J. Mathar, Aug 06 2009

From Ridouane Oudra, Jan 09 2025: (Start)

a(n) = 3*floor((n+1)/3) - n, for n>0.

a(n) = 2*sin(4*Pi*n/3)/sqrt(3), for n>0.

a(n) = - A102283(n), for n>0.

a(n) = - A106510(n), for n>0. (End)

Example

1 - x + x^2 - x^4 + x^5 - x^7 + x^8 - x^10 + x^11 - x^13 + x^14 + ...

Maple

1, seq(2*sin(4*Pi*n/3)/sqrt(3), n=1..100); # Ridouane Oudra, Jan 09 2025

Mathematica

Join[{1}, LinearRecurrence[{-1, -1}, {-1, 1}, 105]] (* Ray Chandler, Sep 15 2015 *)

Prog

(PARI) {a(n) = (n==0) + [0, -1, 1][n%3 + 1]}
(PARI) {a(n) = (n==0) - kronecker(-3, n)}

Crossrefs

A106510(n) = -a(n) unless n=0. Convolution inverse of A117659.

Cf. A102283.

Sequence in context: A106510 A163806 A163810 * A181653 A343159 A155091

Adjacent sequences: A163801 A163802 A163803 * A163805 A163806 A163807

Keyword

sign,easy,changed

Author

Michael Somos, Aug 04 2009

Status

approved