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A052537 Expansion of (1-x)/(1-x-2*x^3).

%I A052537 #36 Jul 02 2025 16:01:58
%S A052537 1,0,0,2,2,2,6,10,14,26,46,74,126,218,366,618,1054,1786,3022,5130,
%T A052537 8702,14746,25006,42410,71902,121914,206734,350538,594366,1007834,
%U A052537 1708910,2897642,4913310,8331130,14126414,23953034,40615294,68868122,116774190,198004778
%N A052537 Expansion of (1-x)/(1-x-2*x^3).
%H A052537 G. C. Greubel, <a href="/A052537/b052537.txt">Table of n, a(n) for n = 0..1000</a>
%H A052537 INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=468">Encyclopedia of Combinatorial Structures 468</a>
%H A052537 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,2).
%F A052537 G.f.: (1-x)/(1 - x - 2*x^3)
%F A052537 a(n) = a(n-1) + 2*a(n-3), with a(0)=1, a(1)=0, a(2)=0.
%F A052537 a(n) = Sum_{alpha = RootOf(-1+x+2*x^3)} (-1/29)*(1 - 10*alpha + 3*alpha^2)*alpha^(-1-n).
%F A052537 a(n) = Sum_{k=1..floor((n-1)/2)} binomial(n-1-2*k, k-1)*2^k, n>=1. - _Taras Goy_, Sep 18 2019
%p A052537 spec := [S,{S=Sequence(Prod(Z,Z,Union(Z,Z),Sequence(Z)))},unlabeled]: seq(combstruct[count](spec,size=n), n=0..20);
%t A052537 CoefficientList[Series[(1-x)/(1-x-2x^3),{x,0,50}],x] (*or*) LinearRecurrence[ {1,0,2},{1,0,0},50] (* _Vladimir Joseph Stephan Orlovsky_, Jan 28 2012 *)
%o A052537 (PARI) my(x='x+O('x^50)); Vec((1-x)/(1-x-2*x^3)) \\ _G. C. Greubel_, May 09 2019
%o A052537 (Magma) R<x>:=PowerSeriesRing(Integers(), 50); Coefficients(R!( (1-x)/(1-x-2*x^3) )); // _G. C. Greubel_, May 09 2019
%o A052537 (Sage) ((1-x)/(1-x-2*x^3)).series(x, 50).coefficients(x, sparse=False) # _G. C. Greubel_, May 09 2019
%o A052537 (GAP) a:=[1,0,0];; for n in [4..50] do a[n]:=a[n-1]+2*a[n-3]; od; a; # _G. C. Greubel_, May 09 2019
%Y A052537 Cf. A003229.
%K A052537 easy,nonn
%O A052537 0,4
%A A052537 encyclopedia(AT)pommard.inria.fr, Jan 25 2000
%E A052537 More terms from _James Sellers_, Jun 05 2000