%I #24 Sep 08 2022 08:44:59
%S 1,0,1,1,2,2,5,5,11,13,25,32,58,78,135,189,316,455,743,1091,1752,2609,
%T 4140,6227,9798,14842,23214,35342,55043,84100,130586,200029,309930,
%U 475601,735789,1130546,1747150,2686951,4149245,6385263,9854895
%N Expansion of (1-x^2)/(1-2*x^2-x^3+x^5).
%H G. C. Greubel, <a href="/A052943/b052943.txt">Table of n, a(n) for n = 0..1000</a>
%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=934">Encyclopedia of Combinatorial Structures 934</a>
%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (0,2,1,0,-1).
%F G.f.: (1-x^2)/(1-2*x^2-x^3+x^5).
%F a(n) = 2*a(n-2) + a(n-3) - a(n-5).
%F a(n) = Sum_{alpha=RootOf(1-2*z^2-z^3+z^5)} (1/4511)*(330 +1089*alpha -224*alpha^2 -167*alpha^3 -100*alpha^4)*alpha^(-1-n).
%p spec := [S,{S=Sequence(Prod(Z,Z,Union(Z,Sequence(Prod(Z,Z)))))},unlabeled ]: seq(combstruct[count ](spec,size=n), n=0..20);
%p seq(coeff(series((1-x^2)/(1-2*x^2-x^3+x^5), x, n+1), x, n), n = 0..50); # _G. C. Greubel_, Oct 18 2019
%t CoefficientList[Series[(1-x^2)/(1-2x^2-x^3+x^5),{x,0,50}],x] (* or *) LinearRecurrence[{0,2,1,0,-1},{1,0,1,1,2},50] (* _Harvey P. Dale_, May 27 2012 *)
%o (PARI) my(x='x+O('x^50)); Vec((1-x^2)/(1-2*x^2-x^3+x^5)) \\ _G. C. Greubel_, Oct 18 2019
%o (Magma) R<x>:=PowerSeriesRing(Integers(), 50); Coefficients(R!( (1-x^2)/(1-2*x^2-x^3+x^5) )); // _G. C. Greubel_, Oct 18 2019
%o (Sage)
%o def A052943_list(prec):
%o P.<x> = PowerSeriesRing(ZZ, prec)
%o return P((1-x^2)/(1-2*x^2-x^3+x^5)).list()
%o A052943_list(50) # _G. C. Greubel_, Oct 18 2019
%o (GAP) a:=[1,0,1,1,2];; for n in [6..50] do a[n]:=2*a[n-2]+a[n-3]-a[n-5]; od; a; # _G. C. Greubel_, Oct 18 2019
%K easy,nonn
%O 0,5
%A encyclopedia(AT)pommard.inria.fr, Jan 25 2000
%E More terms from _James A. Sellers_, Jun 05 2000