OFFSET
0,2
LINKS
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 892
Index entries for linear recurrences with constant coefficients, signature (2,0,2).
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
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 *)
PROG
(PARI) my(x='x+O('x^30)); Vec(1/(1-2*x-2*x^3)) \\ G. C. Greubel, Oct 15 2019
(Magma) R<x>:=PowerSeriesRing(Integers(), 30); Coefficients(R!( 1/(1-2*x-2*x^3) )); // G. C. Greubel, Oct 15 2019
(Sage)
def A052912_list(prec):
P.<x> = PowerSeriesRing(ZZ, prec)
return P(1/(1-2*x-2*x^3)).list()
A052912_list(30) # G. C. Greubel, Oct 15 2019
(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
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
encyclopedia(AT)pommard.inria.fr, Jan 25 2000
EXTENSIONS
More terms from James A. Sellers, Jun 05 2000
STATUS
approved