A052906 - OEIS

%I #56 Jun 01 2024 05:40:48

%S 1,3,9,30,99,327,1080,3567,11781,38910,128511,424443,1401840,4629963,

%T 15291729,50505150,166807179,550926687,1819587240,6009688407,

%U 19848652461,65555645790,216515589831,715102415283,2361822835680

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

%C Image of 1/(1-3*x) under the mapping g(x) -> g(x/(1+x^2)). - _Paul Barry_, Jan 16 2005

%C 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 Robert Israel, <a href="/A052906/b052906.txt">Table of n, a(n) for n = 0..1920</a>

%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=885">Encyclopedia of Combinatorial Structures 885</a>

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (3,1).

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

%F a(n)+3*a(n+1)-a(n+2)=0, with a(0)=1, a(1)=3, a(2)=9.

%F a(n) = Sum_{alpha=RootOf(-1+3*_Z+_Z^2)} (-3/13)*(3*alpha - 2)*alpha^(-1-n).

%F a(n) = Sum_{k=0..floor(n/2)} binomial(n-k-1, k)*3^(n-2*k). - _Paul Barry_, Jan 16 2005

%F 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 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 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 E.g.f.: 1 + 6*exp(3*x/2)*sinh(sqrt(13)*x/2)/sqrt(13). - _Ilya Gutkovskiy_, Nov 11 2016

%F a(n) = A006190(n+1) - A006190(n-1). - _R. J. Mathar_, Oct 06 2017

%F a(n) = 3*Fibonacci(n, 3) for n > 0, where Fibonacci(n, x) are the Fibonacci polynomials. - _G. C. Greubel_, Oct 14 2019

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

%p # alternative:

%p 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 map(f, [$0..50]); # _Robert Israel_, Nov 11 2016

%t Table[If[n==0, 1, 3*Fibonacci[n, 3]], {n,0,30}] (* _G. C. Greubel_, Oct 14 2019 *)

%o (PARI) Vec((1-x^2)/(1-3*x-x^2) + O(x^40)) \\ _Michel Marcus_, Nov 11 2016

%o (Magma) R<x>:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (1-x^2)/(1-3*x-x^2) )); // _G. C. Greubel_, Oct 14 2019

%o (Sage)

%o def A052906_list(prec):

%o     P.<x> = PowerSeriesRing(ZZ, prec)

%o     return P((1-x^2)/(1-3*x-x^2)).list()

%o A052906_list(30) # _G. C. Greubel_, Oct 14 2019

%o (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 Cf. A003688 (partial sums).

%K easy,nonn

%O 0,2

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