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 A189315 #47 Oct 20 2024 01:06:19
%S A189315 5,10,30,100,350,1250,4500,16250,58750,212500,768750,2781250,10062500,
%T A189315 36406250,131718750,476562500,1724218750,6238281250,22570312500,
%U A189315 81660156250,295449218750,1068945312500,3867480468750,13992675781250,50625976562500,183166503906250,662702636718750
%N A189315 Expansion of g.f. 5*(1-3*x+x^2)/(1-5*x+5*x^2).
%C A189315 Let A be the unit-primitive matrix (see [Jeffery])
%C A189315 A=A_(10,1)=
%C A189315 (0 1 0 0 0)
%C A189315 (1 0 1 0 0)
%C A189315 (0 1 0 1 0)
%C A189315 (0 0 1 0 1)
%C A189315 (0 0 0 2 0).
%C A189315 Then a(n) = Trace(A^(2*n)).
%C A189315 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).
%C A189315 From _Tom Copeland_, Dec 08 2015: (Start)
%C A189315 These are also the non-vanishing traces for the adjacency matrices of the simple Lie algebras B_5 and C_5. See links for B_4, A265185, and B_3, A025192.
%C A189315 a(n+1) = 10 * A081567(n), and, ignoring a(0), a G.F. is 10 *(1-2*x)/(1-5*x+5*x^2) whose denominator is y^5 * A127672(5,1/y) with y = sqrt(x).
%C A189315 -log(1 - 5x^2 + 5x^4) = 10 x^2/2 + 30 x^4/4 + ... provides a logarithmic series for the traces of both the odd and even powers of the matrix beginning with the first power. (End)
%H A189315 G. C. Greubel, <a href="/A189315/b189315.txt">Table of n, a(n) for n = 0..1000</a>
%H A189315 L. E. Jeffery, <a href="/wiki/User:L._Edson_Jeffery/Unit-Primitive_Matrices">Unit-primitive matrices</a>.
%H A189315 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (5,-5).
%F A189315 a(n) = 5*a(n-1)-5*a(n-2), n>2, a(0)=5, a(1)=10, a(2)=30.
%F A189315 a(n) = Sum_{k=1..5} (w_k)^(2*n), w_k=2*cos((2*k-1)*Pi/10).
%F A189315 a(n) = 2^(1-n)*((5-Sqrt(5))^n+(5+Sqrt(5))^n), for n>0, with a(0)=5.
%F A189315 a(n) = 5*A147748(n).
%F A189315 E.g.f.: 1 + 4*exp(5*x/2)*cosh(sqrt(5)*x/2). - _Stefano Spezia_, Jul 09 2024
%t A189315 CoefficientList[Series[5(1-3x+x^2)/(1-5x+5x^2),{x,0,40}],x] (* or *)
%t A189315 Join[{5},LinearRecurrence[{5,-5},{10,30},40]] (* _Harvey P. Dale_, Apr 25 2011 *)
%o A189315 (PARI) Vec(5*(1-3*x+x^2)/(1-5*x+5*x^2)+O(x^99)) \\ _Charles R Greathouse IV_, Sep 24 2012
%o A189315 (Magma) I:=[5,10,30]; [n le 3 select I[n] else 5*Self(n-1)-5*Self(n-2): n in [1..30]]; // _Vincenzo Librandi_, Dec 09 2015
%Y A189315 Cf. A147748, A189316, A189317, A189318.
%Y A189315 Cf. A025192, A081567, A127672, A265185.
%K A189315 nonn,easy
%O A189315 0,1
%A A189315 _L. Edson Jeffery_, Apr 20 2011