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A052997 Expansion of (1+x-x^3)/((1-2*x)*(1-x^2)). 3

%I

%S 1,3,7,14,29,58,117,234,469,938,1877,3754,7509,15018,30037,60074,

%T 120149,240298,480597,961194,1922389,3844778,7689557,15379114,

%U 30758229,61516458,123032917,246065834,492131669,984263338,1968526677

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

%H Harvey P. Dale, <a href="/A052997/b052997.txt">Table of n, a(n) for n = 0..1000</a>

%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=1075">Encyclopedia of Combinatorial Structures 1075</a>

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (2,1,-2).

%F G.f.: -(-x+x^3-1)/(-1+x^2)/(-1+2*x).

%F Recurrence: {a(0)=1, -2*a(n)-a(n+1)+a(n+2)-1, a(1)= 3, a(2)=7, a(3)=14}, 11/6*2^n + Sum(-1/6*(2 + _alpha)*_alpha^(-1-n), _alpha=RootOf(-1 + _Z^2))

%F a(n) = 2*a(n-1)+1 for even n, otherwise a(n) = 2*a(n-1), with a(0)=1, a(1)=3. [_Bruno Berselli_, Jun 19 2014]

%F 3*a(n) = 11*2^(n-1)-A000034(n) for n>0. - _R. J. Mathar_, Feb 27 2019

%p spec := [S,{S=Prod(Union(Sequence(Prod(Z,Z)),Z),Sequence(Union(Z,Z)))},unlabeled ]: seq(combstruct[count ](spec,size=n), n=0..20);

%t f[s_List] := Block[{a = s[[-1]]}, Append[s, If[ OddQ@ Length@ s, 2a +1, 2a]]]; Join[{1}, Nest[f, {3}, 30]] (* or *)

%t CoefficientList[ Series[(1 + x - x^3)/(1 - 2x - x^2 + 2x^3), {x, 0, 30}], x] (* _Robert G. Wilson v_, Jul 20 2017 *)

%t LinearRecurrence[{2,1,-2},{1,3,7,14},40] (* _Harvey P. Dale_, May 27 2019 *)

%K nonn,easy

%O 0,2

%A encyclopedia(AT)pommard.inria.fr, Jan 25 2000

%E More terms from _James A. Sellers_, Jun 06 2000

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Last modified July 11 08:50 EDT 2020. Contains 335626 sequences. (Running on oeis4.)