A052537 Expansion of (1-x)/(1-x-2*x^3).

1, 0, 0, 2, 2, 2, 6, 10, 14, 26, 46, 74, 126, 218, 366, 618, 1054, 1786, 3022, 5130, 8702, 14746, 25006, 42410, 71902, 121914, 206734, 350538, 594366, 1007834, 1708910, 2897642, 4913310, 8331130, 14126414, 23953034, 40615294, 68868122, 116774190, 198004778

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 468
• Index entries for linear recurrences with constant coefficients, signature (1,0,2).

Crossrefs

Cf. A003229.

Programs

GAP

a:=[1,0,0];; for n in [4..50] do a[n]:=a[n-1]+2*a[n-3]; od; a; # G. C. Greubel, May 09 2019

Magma

R:=PowerSeriesRing(Integers(), 50); Coefficients(R!( (1-x)/(1-x-2*x^3) )); // G. C. Greubel, May 09 2019

Maple

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

Mathematica

CoefficientList[Series[(1-x)/(1-x-2x^3),{x,0,50}],x] (*or*) LinearRecurrence[ {1,0,2},{1,0,0},50] (* Vladimir Joseph Stephan Orlovsky, Jan 28 2012 *)

PARI

my(x='x+O('x^50)); Vec((1-x)/(1-x-2*x^3)) \\ G. C. Greubel, May 09 2019

Sage

((1-x)/(1-x-2*x^3)).series(x, 50).coefficients(x, sparse=False) # G. C. Greubel, May 09 2019

Formula

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

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

a(n) = Sum_{alpha = RootOf(-1+x+2*x^3)} (-1/29)*(1 - 10*alpha + 3*alpha^2)*alpha^(-1-n).

a(n) = Sum_{k=1..floor((n-1)/2)} binomial(n-1-2*k, k-1)*2^k, n>=1. - Taras Goy, Sep 18 2019

Extensions

More terms from James Sellers, Jun 05 2000