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A052906 Expansion of (1-x^2)/(1-3*x-x^2).

%I A052906 #56 Jun 01 2024 05:40:48
%S A052906 1,3,9,30,99,327,1080,3567,11781,38910,128511,424443,1401840,4629963,
%T A052906 15291729,50505150,166807179,550926687,1819587240,6009688407,
%U A052906 19848652461,65555645790,216515589831,715102415283,2361822835680
%N A052906 Expansion of (1-x^2)/(1-3*x-x^2).
%C A052906 Image of 1/(1-3*x) under the mapping g(x) -> g(x/(1+x^2)). - _Paul Barry_, Jan 16 2005
%C A052906 a(n) is the number of compositions of n into odd parts with three kinds of each part. - _John Tyler Rascoe_, May 28 2024
%H A052906 Robert Israel, <a href="/A052906/b052906.txt">Table of n, a(n) for n = 0..1920</a>
%H A052906 INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=885">Encyclopedia of Combinatorial Structures 885</a>
%H A052906 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (3,1).
%F A052906 G.f.: (1-x^2)/(1-3*x-x^2).
%F A052906 a(n)+3*a(n+1)-a(n+2)=0, with a(0)=1, a(1)=3, a(2)=9.
%F A052906 a(n) = Sum_{alpha=RootOf(-1+3*_Z+_Z^2)} (-3/13)*(3*alpha - 2)*alpha^(-1-n).
%F A052906 a(n) = Sum_{k=0..floor(n/2)} binomial(n-k-1, k)*3^(n-2*k). - _Paul Barry_, Jan 16 2005
%F A052906 If p(i)=3*(i mod 2) and if A is the Hessenberg matrix of order n defined by: A[i,j]=p(j-i+1), (i<=j), A[i,j]=-1, (i=j+1), and A[i,j]=0 otherwise, then, for n>=1, a(n)=det A. - _Milan Janjic_, May 02 2010
%F A052906 G.f.: G(0)*(1-x^2)/(2-3*x), where G(k)= 1 + 1/(1 - (x*(13*k-9))/( x*(13*k+4) - 6/G(k+1))); (continued fraction). - _Sergei N. Gladkovskii_, Jun 15 2013
%F A052906 a(n) = 3*( ((3+sqrt(13))/2)^n - ((3-sqrt(13))/2)^n )/sqrt(13), for n>=1. - _Bogart B. Strauss_, Jul 19 2013
%F A052906 E.g.f.: 1 + 6*exp(3*x/2)*sinh(sqrt(13)*x/2)/sqrt(13). - _Ilya Gutkovskiy_, Nov 11 2016
%F A052906 a(n) = A006190(n+1) - A006190(n-1). - _R. J. Mathar_, Oct 06 2017
%F A052906 a(n) = 3*Fibonacci(n, 3) for n > 0, where Fibonacci(n, x) are the Fibonacci polynomials. - _G. C. Greubel_, Oct 14 2019
%p A052906 spec := [S,{S=Sequence(Prod(Union(Z,Z,Z),Sequence(Prod(Z,Z))))},unlabeled]: seq(combstruct[count](spec,size=n), n=0..20);
%p A052906 # alternative:
%p A052906 f:= gfun:-rectoproc({a(0)=1, a(n)+3*a(n+1)-a(n+2)=0, a(1)=3, a(2)=9},a(n),remember):
%p A052906 map(f, [$0..50]); # _Robert Israel_, Nov 11 2016
%t A052906 Table[If[n==0, 1, 3*Fibonacci[n, 3]], {n,0,30}] (* _G. C. Greubel_, Oct 14 2019 *)
%o A052906 (PARI) Vec((1-x^2)/(1-3*x-x^2) + O(x^40)) \\ _Michel Marcus_, Nov 11 2016
%o A052906 (Magma) R<x>:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (1-x^2)/(1-3*x-x^2) )); // _G. C. Greubel_, Oct 14 2019
%o A052906 (Sage)
%o A052906 def A052906_list(prec):
%o A052906     P.<x> = PowerSeriesRing(ZZ, prec)
%o A052906     return P((1-x^2)/(1-3*x-x^2)).list()
%o A052906 A052906_list(30) # _G. C. Greubel_, Oct 14 2019
%o A052906 (GAP) a:=[3,9];; for n in [3..30] do a[n]:=3*a[n-1]+a[n-2]; od; Concatenation([1], a); # _G. C. Greubel_, Oct 14 2019
%Y A052906 Cf. A003688 (partial sums).
%K A052906 easy,nonn
%O A052906 0,2
%A A052906 encyclopedia(AT)pommard.inria.fr, Jan 25 2000