A052943 Expansion of (1-x^2)/(1-2*x^2-x^3+x^5).

1, 0, 1, 1, 2, 2, 5, 5, 11, 13, 25, 32, 58, 78, 135, 189, 316, 455, 743, 1091, 1752, 2609, 4140, 6227, 9798, 14842, 23214, 35342, 55043, 84100, 130586, 200029, 309930, 475601, 735789, 1130546, 1747150, 2686951, 4149245, 6385263, 9854895

Offset

0,5

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 934

Index entries for linear recurrences with constant coefficients, signature (0,2,1,0,-1).

Formula

G.f.: (1-x^2)/(1-2*x^2-x^3+x^5).

a(n) = 2*a(n-2) + a(n-3) - a(n-5).

a(n) = Sum_{alpha=RootOf(1-2*z^2-z^3+z^5)} (1/4511)*(330 +1089*alpha -224*alpha^2 -167*alpha^3 -100*alpha^4)*alpha^(-1-n).

Maple

spec := [S, {S=Sequence(Prod(Z, Z, Union(Z, Sequence(Prod(Z, Z)))))}, unlabeled ]: seq(combstruct[count ](spec, size=n), n=0..20);
seq(coeff(series((1-x^2)/(1-2*x^2-x^3+x^5), x, n+1), x, n), n = 0..50); # G. C. Greubel, Oct 18 2019

Mathematica

CoefficientList[Series[(1-x^2)/(1-2x^2-x^3+x^5), {x, 0, 50}], x] (* or *) LinearRecurrence[{0, 2, 1, 0, -1}, {1, 0, 1, 1, 2}, 50] (* Harvey P. Dale, May 27 2012 *)

Prog

(PARI) my(x='x+O('x^50)); Vec((1-x^2)/(1-2*x^2-x^3+x^5)) \\ G. C. Greubel, Oct 18 2019
(Magma) R<x>:=PowerSeriesRing(Integers(), 50); Coefficients(R!( (1-x^2)/(1-2*x^2-x^3+x^5) )); // G. C. Greubel, Oct 18 2019
(Sage)
def A052943_list(prec):
    P.<x> = PowerSeriesRing(ZZ, prec)
    return P((1-x^2)/(1-2*x^2-x^3+x^5)).list()
A052943_list(50) # G. C. Greubel, Oct 18 2019
(GAP) a:=[1, 0, 1, 1, 2];; for n in [6..50] do a[n]:=2*a[n-2]+a[n-3]-a[n-5]; od; a; # G. C. Greubel, Oct 18 2019

Crossrefs

Sequence in context: A103891 A265257 A005294 * A325697 A184307 A032580

Adjacent sequences: A052940 A052941 A052942 * A052944 A052945 A052946

Keyword

easy,nonn

Author

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

Extensions

More terms from James A. Sellers, Jun 05 2000

Status

approved