A052910 Expansion of 1 + 2/(1-2*x-x^3).

1, 2, 4, 8, 18, 40, 88, 194, 428, 944, 2082, 4592, 10128, 22338, 49268, 108664, 239666, 528600, 1165864, 2571394, 5671388, 12508640, 27588674, 60848736, 134206112, 296000898, 652850532, 1439907176, 3175815250, 7004481032

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 890
• Index entries for linear recurrences with constant coefficients, signature (2,0,1).

Programs

GAP

a:=[2,4,8];; for n in [4..30] do a[n]:=2*a[n-1]+a[n-3]; od; Concatenation([1], a); # G. C. Greubel, Oct 15 2019

Magma

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

Maple

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

Mathematica

Join[{1},LinearRecurrence[{2,0,1},{2,4,8},30]] (* Harvey P. Dale, Jun 07 2012 *)

PARI

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

Sage

def A052910_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P((1-x^3)/(1-2*x-x^3)).list()
A052910_list(30) # G. C. Greubel, Oct 15 2019

Formula

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

a(n) = 2*a(n-1) + a(n-3), with a(0)=1, a(1)=2, a(2)=4, a(3)=8.

a(n) = Sum_{alpha=RootOf(-1 + 2*z + z^3)} (2/59)*(12 -8*alpha + 9*alpha^2)*alpha^(-1-n).

a(n) = A008998(n) - A008998(n-3). - R. J. Mathar, Nov 28 2011

Extensions

More terms from James Sellers, Jun 05 2000