A128214 Expansion of (1+2x+3x^2)/(1+x+x^2)^2.

1, 0, 0, -2, 3, 0, -5, 6, 0, -8, 9, 0, -11, 12, 0, -14, 15, 0, -17, 18, 0, -20, 21, 0, -23, 24, 0, -26, 27, 0, -29, 30, 0, -32, 33, 0, -35, 36, 0, -38, 39, 0, -41, 42, 0, -44, 45, 0, -47, 48, 0, -50, 51, 0, -53, 54, 0, -56, 57, 0, -59, 60, 0, -62, 63, 0, -65

Offset: 0

Author

Paul Barry, Feb 19 2007

Keywords

• easy
• sign

Comments

Binomial transform is A128213.

Links

• Harvey P. Dale, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (-2,-3,-2,-1).

Crossrefs

Cf. A128213.

Programs

Mathematica

CoefficientList[Series[(1 + 2 x + 3 x^2)/(1 + x + x^2)^2, {x, 0, 50}], x] (* Wesley Ivan Hurt, Mar 15 2015 *)
LinearRecurrence[{-2,-3,-2,-1},{1,0,0,-2},70] (* Harvey P. Dale, Jul 16 2021 *)

PARI

Vec((1+2*x+3*x^2)/(1+x+x^2)^2 + O(x^80)) \\ Michel Marcus, Mar 16 2015

Formula

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

a(n) = (1-n)*cos(2*Pi*n/3)+(n-1)*sin(2*Pi*n/3)/sqrt(3).

a(n) = (-1)^n*( A099254(n)-2*A099254(n-1)+3*A099254(n-2) ). - R. J. Mathar, Mar 21 2011

From Wesley Ivan Hurt, Mar 15 2015: (Start)

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

a(n) = (n-1) * ((n-2)^2 mod 3) * (-1)^floor((2n-2)/3). (End)

Extensions

More terms from Wesley Ivan Hurt, Mar 15 2015