A052906 Expansion of (1-x^2)/(1-3*x-x^2).

1, 3, 9, 30, 99, 327, 1080, 3567, 11781, 38910, 128511, 424443, 1401840, 4629963, 15291729, 50505150, 166807179, 550926687, 1819587240, 6009688407, 19848652461, 65555645790, 216515589831, 715102415283, 2361822835680

Offset: 0

Author

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

Keywords

• easy
• nonn

Comments

Image of 1/(1-3*x) under the mapping g(x) -> g(x/(1+x^2)). - Paul Barry, Jan 16 2005

a(n) is the number of compositions of n into odd parts with three kinds of each part. - John Tyler Rascoe, May 28 2024

Links

• Robert Israel, Table of n, a(n) for n = 0..1920
• INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 885
• Index entries for linear recurrences with constant coefficients, signature (3,1).

Crossrefs

Cf. A003688 (partial sums).

Programs

GAP

a:=[3,9];; for n in [3..30] do a[n]:=3*a[n-1]+a[n-2]; od; Concatenation([1], a); # G. C. Greubel, Oct 14 2019

Magma

R:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (1-x^2)/(1-3*x-x^2) )); // G. C. Greubel, Oct 14 2019

Maple

spec := [S,{S=Sequence(Prod(Union(Z,Z,Z),Sequence(Prod(Z,Z))))},unlabeled]: seq(combstruct[count](spec,size=n), n=0..20);
# alternative:
f:= gfun:-rectoproc({a(0)=1, a(n)+3*a(n+1)-a(n+2)=0, a(1)=3, a(2)=9},a(n),remember):
map(f, [$0..50]); # Robert Israel, Nov 11 2016

Mathematica

Table[If[n==0, 1, 3*Fibonacci[n, 3]], {n,0,30}] (* G. C. Greubel, Oct 14 2019 *)

PARI

Vec((1-x^2)/(1-3*x-x^2) + O(x^40)) \\ Michel Marcus, Nov 11 2016

Sage

def A052906_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P((1-x^2)/(1-3*x-x^2)).list()
A052906_list(30) # G. C. Greubel, Oct 14 2019

Formula

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

a(n)+3*a(n+1)-a(n+2)=0, with a(0)=1, a(1)=3, a(2)=9.

a(n) = Sum_{alpha=RootOf(-1+3*_Z+_Z^2)} (-3/13)*(3*alpha - 2)*alpha^(-1-n).

a(n) = Sum_{k=0..floor(n/2)} binomial(n-k-1, k)*3^(n-2*k). - Paul Barry, Jan 16 2005

If p(i)=3*(i mod 2) and if A is the Hessenberg matrix of order n defined by: A[i,j]=p(j-i+1), (i<=j), A[i,j]=-1, (i=j+1), and A[i,j]=0 otherwise, then, for n>=1, a(n)=det A. - Milan Janjic, May 02 2010

G.f.: G(0)*(1-x^2)/(2-3*x), where G(k)= 1 + 1/(1 - (x*(13*k-9))/( x*(13*k+4) - 6/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 15 2013

a(n) = 3*( ((3+sqrt(13))/2)^n - ((3-sqrt(13))/2)^n )/sqrt(13), for n>=1. - Bogart B. Strauss, Jul 19 2013

E.g.f.: 1 + 6*exp(3*x/2)*sinh(sqrt(13)*x/2)/sqrt(13). - Ilya Gutkovskiy, Nov 11 2016

a(n) = A006190(n+1) - A006190(n-1). - R. J. Mathar, Oct 06 2017

a(n) = 3*Fibonacci(n, 3) for n > 0, where Fibonacci(n, x) are the Fibonacci polynomials. - G. C. Greubel, Oct 14 2019