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A189334 Expansion of g.f. (1-6*x+x^2)/(1-10*x+5*x^2).

%I A189334 #22 Oct 20 2024 01:06:07
%S A189334 1,4,36,340,3220,30500,288900,2736500,25920500,245522500,2325622500,
%T A189334 22028612500,208658012500,1976437062500,18721080562500,
%U A189334 177328620312500,1679680800312500,15910164901562500,150703245014062500,1427481625632812500,13521300031257812500,128075592184414062500
%N A189334 Expansion of g.f. (1-6*x+x^2)/(1-10*x+5*x^2).
%C A189334 (Start) Let A be the unit-primitive matrix (see [Jeffery])
%C A189334 A=A_(10,3)=
%C A189334 (0 0 0 1 0)
%C A189334 (0 0 1 0 1)
%C A189334 (0 1 0 2 0)
%C A189334 (1 0 2 0 1)
%C A189334 (0 2 0 2 0).
%C A189334 Then a(n)=(1/5)*Trace(A^(2*n)). (See also A189317.) (End)
%C A189334 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 A189334 L. E. Jeffery, <a href="/wiki/User:L._Edson_Jeffery/Unit-Primitive_Matrices">Unit-primitive matrices</a>.
%H A189334 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (10,-5).
%F A189334 a(n) = 10*a(n-1) - 5*a(n-2), n>2, a(0)=1, a(1)=4, a(2)=36.
%F A189334 a(n) = (1/5)*Sum_{k=1..5} ((w_k)^3-2*w_k)^(2*n), w_k = 2*cos((2*k-1)*Pi/10).
%F A189334 From _Stefano Spezia_, Jul 09 2024: (Start)
%F A189334 a(n) = 2*((5 - 2*sqrt(5))^n + (5 + 2*sqrt(5))^n)/5 for n > 0.
%F A189334 E.g.f.: (1 + 4*exp(5*x)*cosh(2*sqrt(5)*x))/5. (End)
%t A189334 LinearRecurrence[{10,-5},{1,4,36},22] (* _Stefano Spezia_, Jul 09 2024 *)
%Y A189334 Cf. A189315, A189316, A189317, A189318.
%K A189334 nonn,easy
%O A189334 0,2
%A A189334 _L. Edson Jeffery_, Apr 20 2011
%E A189334 a(20)-a(21) from _Stefano Spezia_, Jul 09 2024