A052912 Expansion of 1/(1-2*x-2*x^3).

1, 2, 4, 10, 24, 56, 132, 312, 736, 1736, 4096, 9664, 22800, 53792, 126912, 299424, 706432, 1666688, 3932224, 9277312, 21888000, 51640448, 121835520, 287447040, 678174976, 1600020992, 3774936064, 8906222080, 21012486144, 49574844416

Offset: 0

Author

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

Keywords

• easy
• nonn

Links

• INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 892
• Index entries for linear recurrences with constant coefficients, signature (2,0,2).

Crossrefs

Cf. A000930.

Programs

GAP

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

Magma

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

Maple

spec := [S,{S=Sequence(Union(Prod(Union(Z,Z),Z,Z),Z,Z))},unlabeled]: seq(combstruct[count](spec,size=n), n=0..20);
seq(coeff(series(1/(1-2*x-2*x^3), x, n+1), x, n), n = 0..30); # G. C. Greubel, Oct 15 2019

Mathematica

LinearRecurrence[{2,0,2}, {1,2,4}, 30] (* G. C. Greubel, Oct 15 2019 *)

PARI

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

Sage

def A052912_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P(1/(1-2*x-2*x^3)).list()
A052912_list(30) # G. C. Greubel, Oct 15 2019

Formula

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

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

a(n) = Sum_{alpha = RootOf(-1 + 2*z + 2*z^3)} (1/43)*(8 + 9*alpha + 12*alpha^2)*alpha^(-1-n).

a(n) = Sum_{k=0..n} binomial(k, floor((n-k)/2)) * 2^k * (1+(-1)^(n-k))/2. - Paul Barry, Jan 12 2006

G.f.: Q(0)/2, where Q(k) = 1 + 1/(1 - x*(4*k+2 + 2*x^2)/( x*(4*k+4 + 2*x^2) + 1/Q(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Aug 30 2013

a(n) = Sum_{k=0..floor(n/3)} binomial(n-2*k, k)*2^(n-2*k). - Greg Dresden, Aug 03 2024

Extensions

More terms from James Sellers, Jun 05 2000