A052911 Expansion of (1-x)/(1 - 3*x - x^2 + 2*x^3).

1, 2, 7, 21, 66, 205, 639, 1990, 6199, 19309, 60146, 187349, 583575, 1817782, 5662223, 17637301, 54938562, 171128541, 533049583, 1660400166, 5171992999, 16110279997, 50182032658, 156312391973, 486898648583, 1516644272406

Offset: 0

Author

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

Keywords

• easy
• nonn

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 891
• Index entries for linear recurrences with constant coefficients, signature (3,1,-2).

Crossrefs

Cf. A100058, A058071, A100059.

Programs

GAP

a:=[1,2,7];; for n in [4..30] do a[n]:=3*a[n-1]+a[n-2]-2*a[n-3]; od; a; # G. C. Greubel, Oct 15 2019

Magma

R:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (1-x)/(1-3*x-x^2+2*x^3) )); // G. C. Greubel, Oct 15 2019

Maple

spec := [S,{S=Sequence(Union(Z,Prod(Union(Sequence(Z),Z,Z),Z)))},unlabeled]: seq(combstruct[count](spec,size=n), n=0..20);

Mathematica

LinearRecurrence[{3,1,-2}, {1,2,7}, 30] (* G. C. Greubel, Oct 15 2019 *)

PARI

my(x='x+O('x^30)); Vec((1-x)/(1-3*x-x^2+2*x^3)) \\ G. C. Greubel, Oct 15 2019

Sage

def A052911_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P((1-x)/(1-3*x-x^2+2*x^3)).list()
A052911_list(30) # G. C. Greubel, Oct 15 2019

Formula

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

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

a(n) = Sum_{alpha=RootOf(1 - 3*z - z^2 + 2*z^3)} (1/229)*(43 + 41*alpha - 46*alpha^2)*alpha^(-1-n).

a(n) = center term in M^n * [1 1 1] where M = Hosoya's triangle considered as an upper triangular 3 X 3 matrix: [2 1 2 / 1 1 0 / 1 0 0]. E.g., a(4) = 66 since M^4 * [1 1 1] = [139 66 45]. The analogous procedure using M^n * [1 0 0] generates A100058. - Gary W. Adamson, Oct 31 2004

a(n) = A100058(n) - A100058(n-1). - R. J. Mathar, May 04 2018

Extensions

More terms from James Sellers, Jun 06 2000