This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A189317 #24 Jun 02 2025 04:01:51
%S A189317 5,20,180,1700,16100,152500,1444500,13682500,129602500,1227612500,
%T A189317 11628112500,110143062500,1043290062500,9882185312500,93605402812500,
%U A189317 886643101562500,8398404001562500,79550824507812500,753516225070312500,7137408128164062500
%N A189317 Expansion of 5*(1-6*x+x^2)/(1-10*x+5*x^2).
%C A189317 (Start) Let A be the unit-primitive matrix (see [Jeffery])
%C A189317 A=A_(10,3)=
%C A189317 (0 0 0 1 0)
%C A189317 (0 0 1 0 1)
%C A189317 (0 1 0 2 0)
%C A189317 (1 0 2 0 1)
%C A189317 (0 2 0 2 0).
%C A189317 Then a(n)=Trace(A^(2*n)). (End)
%C A189317 Evidently one of a class of accelerator sequences for Catalan's constant based on traces of successive powers (here they are A^(2*n)) of a unit-primitive matrix A_(N,r) (0<r<floor(N/2)) and for which the closed-form expression for a(n) is derived from the eigenvalues of A_(N,r).
%H A189317 L. E. Jeffery, <a href="/wiki/User:L._Edson_Jeffery/Unit-Primitive_Matrices">Unit-primitive matrices</a>.
%H A189317 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (10, -5).
%F A189317 G.f.: 5*(1-6*x+x^2)/(1-10*x+5*x^2).
%F A189317 a(n)=10*a(n-1)-5*a(n-2), n>2, a(0)=5, a(1)=20, a(2)=180.
%F A189317 a(n)=Sum_{k=1..5} ((w_k)^3-2*w_k)^(2*n), w_k=2*cos((2*k-1)*Pi/10).
%F A189317 a(n)=2*((5-2*Sqrt(5))^n+(5+2*Sqrt(5))^n), for n>0, with a(0)=5.
%t A189317 CoefficientList[Series[5*(1-6x+x^2)/(1-10x+5x^2),{x,0,30}],x] (* or *) Join[ {5},LinearRecurrence[{10,-5},{20,180},30]] (* _Harvey P. Dale_, Apr 02 2013 *)
%o A189317 (PARI) Vec(5*(1-6*x+x^2)/(1-10*x+5*x^2)+O(x^99)) \\ _Charles R Greathouse IV_, Sep 25 2012
%Y A189317 A189315, A189316, A189318.
%K A189317 nonn,easy
%O A189317 0,1
%A A189317 _L. Edson Jeffery_, Apr 20 2011