A092202 Expansion of (x - x^3) / (1 - x^5) in powers of x.

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

Offset: 0

Author

Paul Barry, Feb 24 2004

Keywords

• easy
• sign

Comments

Partial sums of A080891.

Examples

			x - x^3 + x^6 - x^8 + x^11 - x^13 + x^16 - x^18 + x^21 - x^23 + x^26 + ...
		

Links

• M. E. Muldoon and A. A. Ungar, Beyond Sin and Cos, Mathematics Magazine, 69,1,(1996).
• Index entries for linear recurrences with constant coefficients, signature (-1,-1,-1,-1).

Crossrefs

Cf. A105385 (essentially the same).

Cf. A080891 (first differences), A010891.

Programs

Mathematica

LinearRecurrence[{-1,-1,-1,-1},{0,1,0,-1},120] (* Harvey P. Dale, Jan 14 2022 *)

PARI

{a(n) = (n%5 == 1) - (n%5 == 3)} /* Michael Somos, Mar 26 2012 */

Formula

E.g.f. : F(1, 5, 1, x)-F(1, 5, 3, x);

a(n) = Sum{k=0..n} Jacobi(k, 5).

Euler transform of length 5 sequence [ 0, -1, 0, 0, 1]. - Michael Somos, Mar 26 2012

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

a(n) = a(n + 5). a(-1 - n) = -a(n). a(5*n) = a(5*n + 2) = a(5*n+4) = 0. a(5*n + 1) = 1. a(5*n + 3) = -1. - Michael Somos, Mar 26 2012

a(n) = A010891(n-1)+A010891(n-2). - R. J. Mathar, Aug 11 2021