A052914 Expansion of (1-x)/(1 - x - x^3 - 2*x^4 + 2*x^5).

1, 0, 0, 1, 3, 1, 2, 7, 12, 10, 19, 41, 61, 76, 135, 240, 356, 521, 879, 1445, 2198, 3407, 5568, 8898, 13811, 21797, 35017, 55488, 87111, 138100, 220028, 348081, 549427, 871433, 1383370, 2188903, 3463028, 5490410, 8703187, 13777281, 21815941

Offset

0,5

Links

Harvey P. Dale, Table of n, a(n) for n = 0..1000

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 896

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

Formula

G.f.: (1-x)/(1 - x - x^3 - 2*x^4 + 2*x^5).

a(n) = a(n-1) + a(n-3) + 2*a(n-4) - 2*a(n-5), with a(0)=1, a(1)=0, a(2)=0, a(3)=1, a(4)=3.

a(n) = Sum_{alpha = RootOf(1 - z - z^3 - 2*z^4 + 2*z^5)} (1/19913)*(418 + 4709*alpha - 599*alpha^2 - 1048*alpha^3 - 542*alpha^4)*alpha^(-1-n).

Maple

spec := [S, {S=Sequence(Prod(Union(Sequence(Z), Z, Z), Z, Z, Z))}, unlabeled]: seq(combstruct[count](spec, size=n), n=0..20);
seq(coeff(series((1-x)/(1-x-x^3-2*x^4+2*x^5), x, n+1), x, n), n = 0..50); # G. C. Greubel, Oct 16 2019

Mathematica

CoefficientList[Series[(1-x)/(1-x-x^3-2x^4+2x^5), {x, 0, 50}], x] (* or *) LinearRecurrence[{1, 0, 1, 2, -2}, {1, 0, 0, 1, 3}, 50] (* Harvey P. Dale, Aug 22 2016 *)

Prog

(PARI) my(x='x+O('x^50)); Vec((1-x)/(1-x-x^3-2*x^4+2*x^5)) \\ G. C. Greubel, Oct 16 2019
(Magma) R<x>:=PowerSeriesRing(Integers(), 50); Coefficients(R!( (1-x)/(1-x-x^3-2*x^4+2*x^5) )); // G. C. Greubel, Oct 16 2019
(Sage)
def A052914_list(prec):
    P.<x> = PowerSeriesRing(ZZ, prec)
    return P((1-x)/(1-x-x^3-2*x^4+2*x^5)).list()
A052914_list(50) # G. C. Greubel, Oct 16 2019
(GAP) a:=[1, 0, 0, 1, 3];; for n in [6..50] do a[n]:=a[n-1]+a[n-3]+2*a[n-4] -2*a[n-5]; od; a; # G. C. Greubel, Oct 16 2019

Crossrefs

Sequence in context: A300115 A100888 A322469 * A332491 A144189 A131671

Adjacent sequences: A052911 A052912 A052913 * A052915 A052916 A052917

Keyword

easy,nonn

Author

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

Extensions

More terms from James A. Sellers, May 06 2000

Status

approved