A112300 Expansion of x * (1 - x)^2 * (1 - x^2) / (1 - x^6) in powers of x.

1, -2, 0, 2, -1, 0, 1, -2, 0, 2, -1, 0, 1, -2, 0, 2, -1, 0, 1, -2, 0, 2, -1, 0, 1, -2, 0, 2, -1, 0, 1, -2, 0, 2, -1, 0, 1, -2, 0, 2, -1, 0, 1, -2, 0, 2, -1, 0, 1, -2, 0, 2, -1, 0, 1, -2, 0, 2, -1, 0, 1, -2, 0, 2, -1, 0, 1, -2, 0, 2, -1, 0, 1, -2, 0, 2, -1, 0, 1, -2, 0, 2, -1, 0, 1, -2, 0, 2, -1, 0, 1, -2, 0, 2, -1, 0, 1, -2, 0, 2, -1, 0, 1, -2, 0

Offset: 1

Author

Michael Somos, Sep 02 2005

Keywords

• sign
• easy
• mult

Examples

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

Programs

Mathematica

a[ n_] := {0, 1, -2, 0, 2, -1} [[ Mod[n, 6] + 1]]; (* Michael Somos, May 04 2015 *)
PadRight[{},120,{1,-2,0,2,-1,0}] (* Harvey P. Dale, Jul 09 2019 *)

PARI

{a(n) = [0, 1, -2, 0, 2, -1][n%6 + 1]};

Formula

Euler transform of length 6 sequence [ -2, -1, 0, 0, 0, 1].

Multiplicative with a(2^e) = 2*(-1)^e if e>0, a(3^e) = 0 if e>0, a(p^e) = 1 if p == 1 (mod 6), a(p^e) = (-1)^e if p == 5 (mod 6).

G.f.: x * (1 - x)^2 / ((1 - x + x^2) * (1 + x + x^2)). - Michael Somos, May 04 2015

G.f.: -(f(x) + 3*f(-x)) / 2 where f(x) := x / (1 - x + x^2). - Michael Somos, May 04 2015

a(n) = -a(3 - n) = a(n+6), a(3*n) = 0 for all n in Z.