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A052907 Expansion of 1/(1 - 2*x^2 - 2*x^3).

%I A052907 #69 Jul 02 2025 16:01:58
%S A052907 1,0,2,2,4,8,12,24,40,72,128,224,400,704,1248,2208,3904,6912,12224,
%T A052907 21632,38272,67712,119808,211968,375040,663552,1174016,2077184,
%U A052907 3675136,6502400,11504640,20355072,36014080,63719424,112738304,199467008
%N A052907 Expansion of 1/(1 - 2*x^2 - 2*x^3).
%C A052907 a(n) counts ordered walks of weight n on a single vertex graph containing 4 distinctly labeled loops of weights 2, 2, 3 and 3. - _David Neil McGrath_, Jan 16 2015
%C A052907 Number of compositions (ordered partitions) of n into parts 2 and 3, each of two sorts. - _Joerg Arndt_, Feb 14 2015
%H A052907 G. C. Greubel, <a href="/A052907/b052907.txt">Table of n, a(n) for n = 0..1000</a>
%H A052907 INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=887">Encyclopedia of Combinatorial Structures 887</a>
%H A052907 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (0,2,2).
%F A052907 G.f.: 1/(1 - 2*x^2 - 2*x^3).
%F A052907 a(n) = 2*a(n-2) + 2*a(n-3), with a(0)=1, a(1)=0, a(2)=2.
%F A052907 a(n) = Sum_{alpha = RootOf(-1 + 2*z^2 + 2*z^3)} (-1/19)*(-3 - 5*alpha + 4*alpha^2)*alpha^(-1 - n).
%F A052907 a(n) = Sum_{k=0..floor(n/2)} binomial(k, n-2*k)*2^k. - _Paul Barry_, Oct 19 2004
%F A052907 Construct the matrix T with elements T(n,j) = [A^*j]*[S^*(j-1)](n) with the sequences A = (0,2,2,0,0...) and S = (0,1,0,0...) and the convolution operation *. Define S^*0 = I = (1, repeat(0)). Then T(n,j) for j>=1, counts closed n-walks containing j loops on the graph defined above in a comment, and a(n) = Sum_{j=1..n} T(n,j). - _David Neil McGrath_, Jan 16 2015
%p A052907 spec := [S,{S=Sequence(Prod(Union(Z,Z),Union(Z,Prod(Z,Z))))},unlabeled]: seq(combstruct[count](spec,size=n), n=0..20);
%t A052907 LinearRecurrence[{0,2,2},{1,0,2},40] (* _Harvey P. Dale_, Oct 30 2011 *)
%t A052907 CoefficientList[Series[1/(1-2x^2-2x^3), {x, 0, 40}], x] (* _Vincenzo Librandi_, Feb 14 2015 *)
%o A052907 (Magma) I:=[1,0,2]; [n le 3 select I[n] else 2*Self(n-2)+2*Self(n-3): n in [1..40]]; // _Vincenzo Librandi_, Feb 14 2015
%o A052907 (PARI) my(x='x+O('x^40)); Vec(1/(1-2*x^2-2*x^3)) \\ _G. C. Greubel_, Oct 14 2019
%o A052907 (Sage)
%o A052907 def A052907_list(prec):
%o A052907     P.<x> = PowerSeriesRing(ZZ, prec)
%o A052907     return P( 1/(1-2*x^2-2*x^3) ).list()
%o A052907 A052907_list(40) # _G. C. Greubel_, Oct 14 2019
%o A052907 (GAP) a:=[1,0,2];; for n in [4..40] do a[n]:=2*(a[n-2]+a[n-3]); od; a; # _G. C. Greubel_, Oct 14 2019
%o A052907 (Magma) R<x>:=PowerSeriesRing(Integers(), 36); Coefficients(R!( 1/(1 - 2*x^2 - 2*x^3))); // _Marius A. Burtea_, Oct 15 2019
%K A052907 easy,nonn
%O A052907 0,3
%A A052907 encyclopedia(AT)pommard.inria.fr, Jan 25 2000
%E A052907 More terms from _James Sellers_, Jun 05 2000