cp's OEIS Frontend

A189237 Expansion of (5-12*x-9*x^2+8*x^3+x^4)/(1-3*x-3*x^2+4*x^3+x^4-x^5).

%I A189237 #21 Jun 02 2025 04:01:24
%S A189237 5,3,15,42,155,533,1884,6604,23219,81555,286555,1006734,3537032,
%T A189237 12426742,43659386,153390077,538911123,1893376346,6652069455,
%U A189237 23370962220,82110068595,288480349402,1013528712002,3560868017067,12510529683224
%N A189237 Expansion of (5-12*x-9*x^2+8*x^3+x^4)/(1-3*x-3*x^2+4*x^3+x^4-x^5).
%C A189237 (Start) Let U be the unit-primitive matrix (see [Jeffery])
%C A189237 U=U_(11,4)=
%C A189237 (0 0 0 0 1)
%C A189237 (0 0 0 1 1)
%C A189237 (0 0 1 1 1)
%C A189237 (0 1 1 1 1)
%C A189237 (1 1 1 1 1).
%C A189237 Then a(n)=Trace(U^n). (End)
%C A189237 Evidently one of a class of accelerator sequences for Catalan's constant based on traces of successive powers of a unit-primitive matrix U_(N,r) (0<r<floor(N/2)) and for which the closed-form expression for a(n) is derived from the eigenvalues of U_(N,r).
%H A189237 L. E. Jeffery, <a href="/wiki/User:L._Edson_Jeffery/Unit-Primitive_Matrices">Unit-primitive matrices</a>
%H A189237 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (3, 3, -4, -1, 1).
%F A189237 G.f.: (5-12*x-9*x^2+8*x^3+x^4)/(1-3*x-3*x^2+4*x^3+x^4-x^5).
%F A189237 a(n)=3*a(n-1)+3*a(n-2)-4*a(n-3)-a(n-4)+a(n-5), {a(m)}={5,3,15,42,155}, m=0..4.
%F A189237 a(n)=Sum_{k=1..5} ((x_k)^4-3*(x_k)^2+1)^n; x_k=2*(-1)^(k-1)*cos(k*Pi/11).
%F A189237 Series expansion of g.f. at x=infinity gives -A189234(n+1).
%t A189237 CoefficientList[Series[(5-12x-9x^2+8x^3+x^4)/(1-3x-3x^2+4x^3+x^4-x^5), {x,0,30}],x] (* or *) LinearRecurrence[{3,3,-4,-1,1},{5,3,15,42,155},30] (* _Harvey P. Dale_, Oct 01 2011 *)
%o A189237 (PARI) Vec((5-12*x-9*x^2+8*x^3+x^4)/(1-3*x-3*x^2+4*x^3+x^4-x^5)+O(x^99)) \\ _Charles R Greathouse IV_, Sep 26 2012
%Y A189237 A189234, A189235, A189236.
%K A189237 nonn,easy
%O A189237 0,1
%A A189237 _L. Edson Jeffery_, Apr 18 2011