cp's OEIS Frontend

A052922 Expansion of 1/(1 - 2*x^3 - x^4).

%I A052922 #26 Jul 02 2025 16:01:58
%S A052922 1,0,0,2,1,0,4,4,1,8,12,6,17,32,24,40,81,80,104,202,241,288,508,684,
%T A052922 817,1304,1876,2318,3425,5056,6512,9168,13537,18080,24848,36242,49697,
%U A052922 67776,97332,135636,185249,262440,368604,506134,710129,999648,1380872
%N A052922 Expansion of 1/(1 - 2*x^3 - x^4).
%H A052922 G. C. Greubel, <a href="/A052922/b052922.txt">Table of n, a(n) for n = 0..1000</a>
%H A052922 INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=907">Encyclopedia of Combinatorial Structures 907</a>
%H A052922 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,2,1).
%F A052922 G.f.: 1/(1 - 2*x^3 - x^4).
%F A052922 a(n) = 2*a(n-3) + a(n-4), with a(0)=1, a(1)=0, a(2)=0, a(3)=2.
%F A052922 a(n) = Sum_{alpha=RootOf(-1+2*z^3+z^4)} (1/86)*(4 +26*alpha -3*alpha^2 -6*alpha^3)*alpha^(-1-n).
%p A052922 spec := [S,{S=Sequence(Prod(Z,Z,Union(Z,Z,Prod(Z,Z))))},unlabeled]: seq(combstruct[count](spec,size=n), n=0..50);
%p A052922 seq(coeff(series(1/(1 -2*x^3 -x^4), x, n+1), x, n), n = 0..50); # _G. C. Greubel_, Oct 16 2019
%p A052922 # alternative
%p A052922 A052922 := proc(n)
%p A052922     if n <=3 then
%p A052922         op(n+1,[1,0,0,2]) ;
%p A052922     else
%p A052922         2*procname(n-3)+procname(n-4) ;
%p A052922     end if;
%p A052922 end proc:
%p A052922 seq(A052922(n),n=0..30) ; # _R. J. Mathar_, Nov 22 2024
%t A052922 LinearRecurrence[{0,0,2,1}, {1,0,0,2}, 50] (* _G. C. Greubel_, Oct 16 2019 *)
%t A052922 CoefficientList[Series[1/(1-2x^3-x^4),{x,0,50}],x] (* _Harvey P. Dale_, Nov 04 2024 *)
%o A052922 (PARI) my(x='x+O('x^50)); Vec(1/(1 -2*x^3 -x^4)) \\ _G. C. Greubel_, Oct 16 2019
%o A052922 (Magma) R<x>:=PowerSeriesRing(Integers(), 50); Coefficients(R!( 1/(1 -2*x^3 -x^4) )); // _G. C. Greubel_, Oct 16 2019
%o A052922 (Sage)
%o A052922 def A052922_list(prec):
%o A052922     P.<x> = PowerSeriesRing(ZZ, prec)
%o A052922     return P(1/(1 -2*x^3 -x^4)).list()
%o A052922 A052922_list(50) # _G. C. Greubel_, Oct 16 2019
%o A052922 (GAP) a:=[1,0,0,2];; for n in [5..50] do a[n]:=2*a[n-3]+a[n-4]; od; a; # _G. C. Greubel_, Oct 16 2019
%Y A052922 Cf. A117413 (bijection).
%K A052922 easy,nonn
%O A052922 0,4
%A A052922 encyclopedia(AT)pommard.inria.fr, Jan 25 2000
%E A052922 More terms from _James Sellers_, Jun 05 2000