A052933 Expansion of (1-x^2)/(1 - x - 3*x^2 + 2*x^4).

1, 1, 3, 6, 13, 29, 62, 137, 297, 650, 1417, 3093, 6750, 14729, 32145, 70146, 153081, 334061, 729014, 1590905, 3471785, 7576378, 16533705, 36081029, 78738574, 171828905, 374977217, 818301874, 1785756377, 3897004189, 8504318886

Offset

0,3

Comments

Starting (1, 3, 6, ...) equals INVERT transform of (1, 2, 1, 0, 1, 0, 1, ...). - Gary W. Adamson, Apr 28 2009

Links

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

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 920

Index entries for linear recurrences with constant coefficients, signature (1,3,0,-2).

Formula

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

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

a(n) = Sum_{alpha=RootOf(1 - z - 3*z^2 + 2*z^4)} (1/397)*(51 + 148*alpha - 27*alpha^2 - 82*alpha^3)*alpha^(-1-n).

Maple

spec:= [S, {S=Sequence(Prod(Z, Union(Z, Z, Sequence(Prod(Z, Z)))))}, unlabeled ]: seq(combstruct[count ](spec, size=n), n=0..20);
seq(coeff(series((1-x^2)/(1-x-3*x^2+2*x^4), x, n+1), x, n), n = 0..40); # G. C. Greubel, Oct 18 2019

Mathematica

CoefficientList[Series[(1-x^2)/(1-x-3x^2+2x^4), {x, 0, 40}], x] (* or *) LinearRecurrence[{1, 3, 0, -2}, {1, 1, 3, 6}, 40] (* Harvey P. Dale, Mar 23 2012 *)

Prog

(PARI) my(x='x+O('x^40)); Vec((1-x^2)/(1-x-3*x^2+2*x^4)) \\ G. C. Greubel, Oct 18 2019
(Magma) R<x>:=PowerSeriesRing(Integers(), 40); Coefficients(R!( (1-x^2)/(1-x-3*x^2+2*x^4) )); // G. C. Greubel, Oct 18 2019
(Sage)
def A052933_list(prec):
    P.<x> = PowerSeriesRing(ZZ, prec)
    return P((1-x^2)/(1-x-3*x^2+2*x^4)).list()
A052933_list(40) # G. C. Greubel, Oct 18 2019
(GAP) a:=[1, 1, 3, 6];; for n in [5..40] do a[n]:=a[n-1]+3*a[n-2]-2*a[n-4]; od; a; # G. C. Greubel, Oct 18 2019

Crossrefs

Sequence in context: A095768 A002478 A106496 * A071014 A214260 A078061

Adjacent sequences: A052930 A052931 A052932 * A052934 A052935 A052936

Keyword

easy,nonn

Author

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

Extensions

More terms from James A. Sellers, Jun 06 2000

Status

approved