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

%I A052910 #25 Jul 02 2025 16:01:58
%S A052910 1,2,4,8,18,40,88,194,428,944,2082,4592,10128,22338,49268,108664,
%T A052910 239666,528600,1165864,2571394,5671388,12508640,27588674,60848736,
%U A052910 134206112,296000898,652850532,1439907176,3175815250,7004481032
%N A052910 Expansion of 1 + 2/(1-2*x-x^3).
%H A052910 G. C. Greubel, <a href="/A052910/b052910.txt">Table of n, a(n) for n = 0..1000</a>
%H A052910 INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=890">Encyclopedia of Combinatorial Structures 890</a>
%H A052910 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (2,0,1).
%F A052910 G.f.: (1-x^3)/(1-2*x-x^3).
%F A052910 a(n) = 2*a(n-1) + a(n-3), with a(0)=1, a(1)=2, a(2)=4, a(3)=8.
%F A052910 a(n) = Sum_{alpha=RootOf(-1 + 2*z + z^3)} (2/59)*(12 -8*alpha + 9*alpha^2)*alpha^(-1-n).
%F A052910 a(n) = A008998(n) - A008998(n-3). - _R. J. Mathar_, Nov 28 2011
%p A052910 spec := [S,{S=Sequence(Prod(Sequence(Prod(Z,Z,Z)),Union(Z,Z)))},unlabeled]: seq(combstruct[count](spec,size=n), n=0..20);
%t A052910 Join[{1},LinearRecurrence[{2,0,1},{2,4,8},30]] (* _Harvey P. Dale_, Jun 07 2012 *)
%o A052910 (PARI) my(x='x+O('x^30)); Vec((1-x^3)/(1-2*x-x^3)) \\ _G. C. Greubel_, Oct 15 2019
%o A052910 (Magma) R<x>:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (1-x^3)/(1-2*x-x^3) )); // _G. C. Greubel_, Oct 15 2019
%o A052910 (Sage)
%o A052910 def A052910_list(prec):
%o A052910     P.<x> = PowerSeriesRing(ZZ, prec)
%o A052910     return P((1-x^3)/(1-2*x-x^3)).list()
%o A052910 A052910_list(30) # _G. C. Greubel_, Oct 15 2019
%o A052910 (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
%K A052910 easy,nonn
%O A052910 0,2
%A A052910 encyclopedia(AT)pommard.inria.fr, Jan 25 2000
%E A052910 More terms from _James Sellers_, Jun 05 2000