A163811 Expansion of (1 - x) * (1 - x^10) / ((1 - x^5) * (1 - x^6)) in powers of x.

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

Offset: 0

Author

Michael Somos, Aug 04 2009, Aug 09 2009

Keywords

• sign
• easy

Examples

			1 - x + x^5 - x^7 + x^11 - x^13 + x^17 - x^19 + x^23 - x^25 + x^29 + ...
		

Links

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

Crossrefs

A163817(n) = -a(n) unless n=0. A163817(n) = (-1)^n * a(n).

Convolution inverse of A163812.

Programs

Mathematica

Join[{1}, LinearRecurrence[{0, -1, 0, -1}, {-1, 0, 0, 0}, 50]] (* G. C. Greubel, Aug 04 2017 *)

PARI

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

PARI

{a(n) = (n==0) - kronecker(-12, n)}

Formula

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

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

a(-n) = -a(n) unless n=0. a(n+6) = a(n) unless n=0 or n=-6.

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

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