A112553 Expansion of 1/( (1+x^2)*(1-x+x^2) ).

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

Offset: 0

Author

Paul Barry, Sep 13 2005

Keywords

• easy
• sign

Comments

Row sums of A112552.

Examples

			G.f. = 1 + x - x^2 - 2*x^3 + 2*x^5 + x^6 - x^7 - x^8 + x^12 + x^13 - x^14 - 2*x^15 + ... - _Michael Somos_, Feb 15 2024
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (1,-2,1,-1).

Crossrefs

Cf. A010892, A112552, A112554.

Programs

Mathematica

Join[{1}, PadRight[{}, 120, {1,-1,-2,0,2,1,-1,-1,0,0,0,1}]] (* G. C. Greubel, Jan 13 2022 *)
a[ n_] := {1, 1, -1, -2, 0, 2, 1, -1, -1, 0, 0, 0}[[1 + Mod[n, 12]]]; (* Michael Somos, Feb 15 2024 *)

PARI

{a(n) = [1, 1, -1, -2, 0, 2, 1, -1, -1, 0, 0, 0][1 + n%12]}; /* Michael Somos, Feb 15 2024 */

Formula

a(n) = (1/4)*Sum_{k=0..n} (-1)^floor((n-k)/2)*Sum_{j=0..n} (1+(-1)^(n-j))*(1+(-1)^(j-k)) *binomial((j+k)/2, k).

From G. C. Greubel, Jan 13 2022: (Start)

a(n) = Sum_{k=0..n} (-1)^floor((n-k)/2)*((1 + (-1)^(n+k))/2)*binomial((n+k+2)/2, k+1).

a(n + 12) = a(n). (End)

a(n) = -a(-4-n) for all n in Z. - Michael Somos, Feb 15 2024