A163812 Expansion of (1 - x^5) * (1 - x^6) / ((1 - x) * (1 - x^10)) in powers of x.

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

Offset: 0

Author

Michael Somos, Aug 04 2009

Keywords

• sign
• easy

Examples

			G.f. = 1 + x + x^2 + x^3 + x^4 - x^6 - x^7 - x^8 - x^9 + x^11 + x^12 + x^13 + ...
		

Links

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

Crossrefs

Cf. A099443, A163811, A163818.

Programs

Mathematica

a[ n_] := Boole[n == 0] + (-1)^Quotient[n, 5] Sign@Mod[n, 5]; (* Michael Somos, Jun 17 2015 *)

PARI

{a(n) = (n==0) + [0, 1, 1, 1, 1, 0, -1, -1, -1, -1][n%10 + 1]};

Formula

Euler transform of length 10 sequence [ 1, 0, 0, 0, -1, -1, 0, 0, 0, 1].

a(5*n) = 0 unless n=0.

a(n) = -a(-n) unless n=0. a(n+5) = -a(n) unless n=0 or n=-5.

G.f.: (1 + x^2 + x^4) / (1 - x + x^2 - x^3 + x^4).

a(n) = (-1)^n * A163818(n). Convolution inverse of A163811.

G.f.: A(x) = 1 / (1 - x / ( 1 + x^4 / (1 + x^2))) = 1 + x / (1 - x / (1 + x^3 / (1 + x^2 / (1 + x / (1 - x))))). - Michael Somos, Jan 03 2013

a(n) = A099443(n-1), n>0. - R. J. Mathar, Aug 05 2009