A163810 Expansion of (1 - x) * (1 - x^2) * (1 - x^3) / (1 - x^6) 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

Offset: 0

Author

Michael Somos, Nov 07 2007

Keywords

• sign
• easy

Examples

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

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (1,-1).

Crossrefs

A163806(n) = -a(n) unless n=0. A106510(n) = (-1)^n * a(n).

Convolution inverse of A028310. Series reversion of A109081.

Programs

Mathematica

Join[{1},LinearRecurrence[{1, -1},{-1, -1},104]] (* Ray Chandler, Sep 15 2015 *)

PARI

{a(n) = (n==0) + [0, -1, -1, 0, 1, 1][n%6 + 1]};

PARI

{a(n) = (n==0) + (-1)^n * kronecker(-3, n)};

Formula

Euler transform of length 6 sequence [ -1, -1, -1, 0, 0, 1].

G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = 2 * u * (1 - u) * (2 - v) - (v - u^2).

a(3*n) = 0 unless n=0. a(6*n + 1) = a(6*n + 2) = -1, a(6*n + 4) = a(6*n + 5) = 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).