A052947 Expansion of 1/(1-x^2-2*x^3).

1, 0, 1, 2, 1, 4, 5, 6, 13, 16, 25, 42, 57, 92, 141, 206, 325, 488, 737, 1138, 1713, 2612, 3989, 6038, 9213, 14016, 21289, 32442, 49321, 75020, 114205, 173662, 264245, 402072, 611569, 930562, 1415713, 2153700, 3276837, 4985126, 7584237, 11538800

Offset: 0

Author

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

Keywords

• easy
• nonn

Comments

a(n) is the number of compositions (ordered partitions) of n into parts 1 of one kind and part 2 of two kinds. - Joerg Arndt, Oct 22 2019

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 1006
• Index entries for linear recurrences with constant coefficients, signature (0,1,2).

Crossrefs

Column k=2 of A219946. - Alois P. Heinz, Dec 01 2012

Programs

GAP

a:=[1,0,1];; for n in [4..50] do a[n]:=a[n-2]+2*a[n-3]; od; a; # G. C. Greubel, Oct 21 2019

Magma

R:=PowerSeriesRing(Integers(), 50); Coefficients(R!( 1/(1-x^2-2*x^3) )); // G. C. Greubel, Oct 21 2019

Maple

spec:= [S,{S=Sequence(Prod(Union(Prod(Union(Z,Z),Z),Z),Z))}, unlabeled]: seq(combstruct[count ](spec,size=n), n=0..20);
seq(coeff(series(1/(1-x^2-2*x^3), x, n+1), x, n), n = 0..50); # G. C. Greubel, Oct 21 2019

Mathematica

LinearRecurrence[{0,1,2}, {1,0,1}, 50] (* G. C. Greubel, Oct 21 2019 *)

PARI

my(x='x+O('x^50)); Vec(1/(1-x^2-2*x^3)) \\ G. C. Greubel, Oct 21 2019

Sage

def A052947_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P(1/(1-x^2-2*x^3)).list()
A052947_list(50) # G. C. Greubel, Oct 21 2019

Formula

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

a(n) = Sum_{alpha=RootOf(-1+z^2+2*z^3)} (1/52)*(3 +17*alpha -2*alpha^2)*alpha^(-1-n).

a(n) = Sum_{k=0..floor(n/2)} C(k,n-2*k)*2^(n-2*k). - Paul Barry, Oct 16 2004

If p[1]=0, p[2]=1, p[3]=2, p[i]=0, (i>3), and if A is 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-1)=det A. - Milan Janjic, May 02 2010

Extensions

More terms from James Sellers, Jun 05 2000