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 A189316 #33 Oct 20 2024 01:06:15
%S A189316 5,5,15,35,95,245,645,1685,4415,11555,30255,79205,207365,542885,
%T A189316 1421295,3720995,9741695,25504085,66770565,174807605,457652255,
%U A189316 1198149155,3136795215,8212236485,21499914245,56287506245,147362604495,385800307235,1010038317215
%N A189316 Expansion of g.f. 5*(1-x-x^2)/((1+x)*(1-3*x+x^2)).
%C A189316 (Start) Let A be the unit-primitive matrix (see [Jeffery])
%C A189316 A=A_(10,2)=
%C A189316 (0 0 1 0 0)
%C A189316 (0 1 0 1 0)
%C A189316 (1 0 1 0 1)
%C A189316 (0 1 0 2 0)
%C A189316 (0 0 2 0 1).
%C A189316 Then a(n)=Trace(A^n). For m=1,2,..., A^(m) can also be written
%C A189316 A^(m)=
%C A189316 [ F(m-1)^2 0 F(m)^2 0 F(m-1)*F(m) ]
%C A189316 [ 0 F(2*m-1) 0 F(2*m) 0 ]
%C A189316 [ F(m)^2 0 F(m+1)^2 0 F(m)*F(m+1) ]
%C A189316 [ 0 F(2*m) 0 F(2*m+1) 0 ]
%C A189316 [ 2*F(m-1)*F(m) 0 2*F(m)*F(m+1) 0 F(2*m+1)-F(m)*F(m+1) ],
%C A189316 where F(m-1)=A000045(n) are the Fibonacci numbers and m=n+1. Hence also a(n+1)=Trace(A^(n+1))=F(m-1)^2+F(2*m-1)+F(m+1)^2+2*F(2*m+1)-F(m)*F(m+1). (End)
%C A189316 Evidently one of a class of accelerator sequences for Catalan's constant based on traces of successive powers 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 A189316 L. E. Jeffery, <a href="/wiki/User:L._Edson_Jeffery/Unit-Primitive_Matrices">Unit-primitive matrices</a>.
%H A189316 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (2,2,-1).
%F A189316 G.f.: 5*(1-x-x^2)/((1+x)*(1-3*x+x^2)).
%F A189316 a(n) = 2*a(n-1)+2*a(n-2)-a(n-3), n>2, a(0)=5, a(1)=5, a(2)=15.
%F A189316 a(n) = Sum_{k=1..5} ((w_k)^2-1)^n, w_k = 2*cos((2*k-1)*Pi/10).
%F A189316 a(n) = (-1)^n+2*(1/tau^(2*n)+tau^(2*n)), tau = (1+sqrt(5))/2=1.618033....
%F A189316 a(n) = 5*A061646(n), n>=0 (offset for A061646 is -1).
%F A189316 E.g.f.: cosh(x) + 4*exp(3*x/2)*cosh(sqrt(5)*x/2) - sinh(x). - _Stefano Spezia_, Jul 09 2024
%t A189316 CoefficientList[Series[5 (1-x-x^2)/((1+x)(1-3x+x^2)),{x,0,40}],x] (* or *) LinearRecurrence[{2,2,-1},{5,5,15},40] (* _Harvey P. Dale_, Nov 26 2016 *)
%Y A189316 Cf. A000045, A061646, A189315, A189317, A189318.
%K A189316 nonn,easy
%O A189316 0,1
%A A189316 _L. Edson Jeffery_, Apr 20 2011