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A052987 Expansion of (1-2x^2)/(1-2x-2x^2+2x^3).

%I A052987 #29 Jul 02 2025 16:01:58
%S A052987 1,2,4,10,24,60,148,368,912,2264,5616,13936,34576,85792,212864,528160,
%T A052987 1310464,3251520,8067648,20017408,49667072,123233664,305766656,
%U A052987 758666496,1882398976,4670597632,11588660224,28753717760,71343560704
%N A052987 Expansion of (1-2x^2)/(1-2x-2x^2+2x^3).
%C A052987 Form the graph with matrix A=[1,1,1,1;1,0,0,0;1,0,0,0;1,0,0,1]. Then the sequence 1,1,2,4,... with g.f. (1-x-2x^2)/(1-2x-2x^2+2x^3) counts closed walks of length n at the degree 3 vertex. - _Paul Barry_, Oct 02 2004
%C A052987 Equals INVERT transform of the Jacobsthal sequence A001045 prefaced with a 1:
%C A052987 [1, 1, 1, 3, 5, 11, 21, 43, ...]. - _Gary W. Adamson_, May 27 2009
%H A052987 INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=1061">Encyclopedia of Combinatorial Structures 1061</a>
%H A052987 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (2,2,-2).
%F A052987 G.f.: -(-1+2*x^2)/(1-2*x-2*x^2+2*x^3)
%F A052987 Recurrence: {a(0)=1, a(2)=4, a(1)=2, 2*a(n)-2*a(n+1)-2*a(n+2)+a(n+3)=0}
%F A052987 Sum(1/37*(6+7*_alpha+4*_alpha^2)*_alpha^(-1-n), _alpha=RootOf(2*_Z^3-2*_Z^2-2*_Z+1)).
%p A052987 spec := [S,{S=Sequence(Union(Prod(Sequence(Prod(Union(Z,Z),Z)),Z),Z))},unlabeled ]: seq(combstruct[count ](spec,size=n), n=0..20);
%t A052987 InvertTransform[ser_, n_] := CoefficientList[ Series[1/(1 - x ser), {x,0,n}],x];
%t A052987 Jacobsthal := (2x^2-1)/((x + 1)(2x - 1));
%t A052987 PadLeft[InvertTransform[Jacobsthal, 29],29,1] (* _Peter Luschny_, Jan 10 2019 *)
%Y A052987 Cf. A077847, A052528, A077937, A001045.
%K A052987 easy,nonn
%O A052987 0,2
%A A052987 encyclopedia(AT)pommard.inria.fr, Jan 25 2000
%E A052987 More terms from _James Sellers_, Jun 05 2000