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%I #28 Sep 08 2022 08:44:59
%S 1,2,4,10,24,56,132,312,736,1736,4096,9664,22800,53792,126912,299424,
%T 706432,1666688,3932224,9277312,21888000,51640448,121835520,287447040,
%U 678174976,1600020992,3774936064,8906222080,21012486144,49574844416
%N Expansion of 1/(1-2*x-2*x^3).
%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=892">Encyclopedia of Combinatorial Structures 892</a>
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (2,0,2).
%F G.f.: 1/(1 - 2*x - 2*x^3)
%F a(n) = 2*a(n-1) +2*a(n-3).
%F a(n) = Sum_{alpha = RootOf(-1 + 2*z + 2*z^3)} (1/43)*(8 + 9*alpha + 12*alpha^2)*alpha^(-1-n).
%F a(n) = Sum_{k=0..n} binomial(k, floor((n-k)/2)) * 2^k * (1+(-1)^(n-k))/2. - _Paul Barry_, Jan 12 2006
%F 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
%p spec := [S,{S=Sequence(Union(Prod(Union(Z,Z),Z,Z),Z,Z))},unlabeled]: seq(combstruct[count](spec,size=n), n=0..20);
%p seq(coeff(series(1/(1-2*x-2*x^3), x, n+1), x, n), n = 0..30); # _G. C. Greubel_, Oct 15 2019
%t LinearRecurrence[{2,0,2}, {1,2,4}, 30] (* _G. C. Greubel_, Oct 15 2019 *)
%o (PARI) my(x='x+O('x^30)); Vec(1/(1-2*x-2*x^3)) \\ _G. C. Greubel_, Oct 15 2019
%o (Magma) R<x>:=PowerSeriesRing(Integers(), 30); Coefficients(R!( 1/(1-2*x-2*x^3) )); // _G. C. Greubel_, Oct 15 2019
%o (Sage)
%o def A052912_list(prec):
%o P.<x> = PowerSeriesRing(ZZ, prec)
%o return P(1/(1-2*x-2*x^3)).list()
%o A052912_list(30) # _G. C. Greubel_, Oct 15 2019
%o (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
%K easy,nonn
%O 0,2
%A encyclopedia(AT)pommard.inria.fr, Jan 25 2000
%E More terms from _James A. Sellers_, Jun 05 2000
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