A109247 Expansion of (1 - 3*x^2 - 3*x^3 + x^4)/(1 + x^4).

1, 0, -3, -3, 0, 0, 3, 3, 0, 0, -3, -3, 0, 0, 3, 3, 0, 0, -3, -3, 0, 0, 3, 3, 0, 0, -3, -3, 0, 0, 3, 3, 0, 0, -3, -3, 0, 0, 3, 3, 0, 0, -3, -3, 0, 0, 3, 3, 0, 0, -3, -3, 0, 0, 3, 3, 0, 0, -3, -3, 0, 0, 3, 3, 0, 0, -3, -3, 0, 0, 3, 3, 0, 0, -3, -3, 0, 0, 3, 3, 0, 0, -3, -3

Offset: 0

Author

Paul Barry, Jun 23 2005

Keywords

• sign
• easy

Comments

Row sums of Riordan array (1-x-2x^2,x(1-x)), A109246.

After the initial terms, cyclic with period 8: [0,0,-3,-3,0,0,3,3]. - Antti Karttunen, Aug 12 2017

Links

• Antti Karttunen, Table of n, a(n) for n = 0..8191
• Index entries for linear recurrences with constant coefficients, signature (0,0,0,-1).

Crossrefs

Cf. A109246, A132380.

Programs

Mathematica

CoefficientList[Series[(1-3x^2-3x^3+x^4)/(1+x^4),{x,0,90}],x] (* or *) Join[{1},LinearRecurrence[{0,0,0,-1},{0,-3,-3,0},90]] (* Harvey P. Dale, Mar 24 2012 *)

PARI

Vec((1-3*x^2-3*x^3+x^4)/(1+x^4) + O(x^80)) \\ Jinyuan Wang, Mar 22 2020

Scheme

(define (A109247 n) (case n ((0) 1) ((1 4) 0) ((2 3) -3) (else (- (A109247 (- n 4)))))) ;; (After Harvey P. Dale's Mar 24 2012 recurrence) - Antti Karttunen, Aug 12 2017

Formula

a(0)=1, a(1)=0, a(2)=-3, a(3)=-3, a(4)=0, a(n)=-a(n-4) - Harvey P. Dale, Mar 24 2012

For n > 0, a(n) = -3 * A132380(n). - Antti Karttunen, Aug 12 2017