A052915 Expansion of (1-x)/(1 - x - x^2 - 3*x^3 + 3*x^4).

1, 0, 1, 4, 2, 9, 20, 23, 64, 120, 193, 436, 797, 1452, 2978, 5513, 10456, 20547, 38608, 73984, 142865, 271032, 520025, 997700, 1902226, 3646905, 6982156, 13342639, 25558832, 48907224, 93547505, 179103308, 342695989, 655720140, 1255083538

Offset

0,4

Links

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

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 897

Index entries for linear recurrences with constant coefficients, signature (1,1,3,-3).

Formula

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

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

a(n) = Sum_{alpha=RootOf(1 - z - z^2 - 3*z^3 + 3*z^4)} (1/2857)*(142 + 885*alpha - 240*alpha^2 - 351*alpha^3)*alpha^(-1-n).

Maple

spec := [S, {S=Sequence(Prod(Z, Z, Union(Sequence(Z), Z, Z, Z)))}, unlabeled]: seq(combstruct[count](spec, size=n), n=0..20);
seq(coeff(series((1-x)/(1-x-x^2-3*x^3+3*x^4), x, n+1), x, n), n = 0..40); # G. C. Greubel, Oct 16 2019

Mathematica

LinearRecurrence[{1, 1, 3, -3}, {1, 0, 1, 4}, 40] (* G. C. Greubel, Oct 16 2019 *)

Prog

(PARI) my(x='x+O('x^40)); Vec((1-x)/(1-x-x^2-3*x^3+3*x^4)) \\ G. C. Greubel, Oct 16 2019
(Magma) R<x>:=PowerSeriesRing(Integers(), 40); Coefficients(R!( (1-x)/(1-x-x^2-3*x^3+3*x^4) )); // G. C. Greubel, Oct 16 2019
(Sage)
def A052915_list(prec):
    P.<x> = PowerSeriesRing(ZZ, prec)
    return P((1-x)/(1-x-x^2-3*x^3+3*x^4)).list()
A052915_list(40) # G. C. Greubel, Oct 16 2019
(GAP) a:=[1, 0, 1, 4];; for n in [5..40] do a[n]:=a[n-1]+a[n-2]+3*a[n-3] -3*a[n-4]; od; a; # G. C. Greubel, Oct 16 2019

Crossrefs

Sequence in context: A298567 A006172 A171631 * A130273 A016516 A349693

Adjacent sequences: A052912 A052913 A052914 * A052916 A052917 A052918

Keyword

easy,nonn

Author

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

Extensions

More terms from James A. Sellers, Jun 06 2000

Status

approved