A105385 Expansion of (1-x^2)/(1-x^5).

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

Offset: 0

Author

Paul Barry, Apr 02 2005

Keywords

• sign
• easy

Comments

Periodic {1,0,-1,0,0}.

Binomial transform is A103311(n+1). Consecutive pair sums of A105384.

Links

• Index entries for linear recurrences with constant coefficients, signature (-1,-1,-1,-1).

Crossrefs

Cf. A092202 (essentially the same).

Cf. A198517 (absolute values).

Cf. A103311, A105384

Programs

Mathematica

CoefficientList[Series[(1-x^2)/(1-x^5),{x,0,100}],x] (* or *) PadRight[{},100,{1,0,-1,0,0}] (* or *) LinearRecurrence[{-1,-1,-1,-1},{1,0,-1,0},100] (* Harvey P. Dale, Mar 10 2013 *)

Formula

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

a(n) = sqrt(1/5 - 2*sqrt(5)/25)*cos(4*Pi*n/5 + Pi/10) + sqrt(5)*sin(4*Pi*n/5 + Pi/10)/5 + sqrt(2*sqrt(5)/25 + 1/5)*cos(2*Pi*n/5 + 3*Pi/10) + sqrt(5)*sin(2*Pi*n/5 + 3*Pi/10)/5.

a(n) = A092202(n+1). - R. J. Mathar, Aug 28 2008

a(n) = a(n-1) - a(n-2) - a(n-3) - a(n-4); a(0)=1, a(1)=0, a(2)=-1, a(3)=0. - Harvey P. Dale, Mar 10 2013