A163818 Expansion of (1 - x) * (1 - x^6) / ((1 - x^2) * (1 - x^5)) 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. A163813, A163817.

Programs

Mathematica

a[ n_] := Boole[n == 0] + {-1, 1, -1, 1, 0}[[Mod[n, 5, 1]]]; (* Michael Somos, Jun 17 2015 *)
a[ n_] := Boole[n == 0] + (-1)^(n + Quotient[n, 5]) Sign@Mod[n, 5]; (* Michael Somos, Jun 17 2015 *)
CoefficientList[Series[(1 + x^2 + x^4) / (1 + x + x^2 + x^3 + x^4), {x, 0, 100}], x] (* Vincenzo Librandi, Aug 05 2017 *)

PARI

{a(n) = (n==0) + [ 0, -1, 1, -1, 1][n%5 + 1]};

PARI

{a(n) = (n==0) + (-1)^(n + n\5) * kronecker(25, n)};

Formula

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

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

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).

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

a(n) = (-1)^n * A163812(n). Convolution inverse of A163817.