%I #22 Aug 16 2015 12:03:59

%S 5,5,25,65,225,705,2305,7425,24065,77825,251905,815105,2637825,

%T 8536065,27623425,89391105,289275905,936116225,3029336065,9803137025,

%U 31723618305,102659784705,332214042625,1075067224065,3478990618625,11258250133505

%N Expansion of 5*(1-2*x)/(1-3*x-2*x^2+4*x^3)

%C (Start) Let A be the unit-primitive matrix (see [Jeffery])

%C A=A_(10,4)=

%C (0 0 0 0 1)

%C (0 0 0 2 0)

%C (0 0 2 0 1)

%C (0 2 0 2 0)

%C (2 0 2 0 1).

%C Then a(n)=Trace(A^n). (End)

%C 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 L. E. Jeffery, <a href="/wiki/User:L._Edson_Jeffery/Unit-Primitive_Matrices">Unit-primitive matrices</a>.

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3, 2, -4).

%F G.f.: 5*(1-2*x)/(1-3*x-2*x^2+4*x^3).

%F a(n)=3*a(n-1)+2*a(n-2)-4*a(n-3), n>3, a(0)=5, a(1)=5, a(2)=25, a(3)=65.

%F a(n)=Sum_{k=1..5} ((w_k)^4-3*(w_k)^2+1)^n, w_k=2*cos((2*k-1)*Pi/10).

%F a(n)=1+2*(1-Sqrt(5))^n+2*(1+Sqrt(5))^n.

%F a(n)=5*A052899(n).

%t CoefficientList[Series[5(1-2x)/(1-3x-2x^2+4x^3),{x,0,30}],x] (* or *) LinearRecurrence[{3,2,-4},{5,5,25},30] (* _Harvey P. Dale_, Jun 02 2014 *)

%o (PARI) Vec(5*(1-2*x)/(1-3*x-2*x^2+4*x^3)+O(x^99)) \\ _Charles R Greathouse IV_, Sep 25 2012

%Y Cf. A052899.

%Y A189315, A189316, A189317.

%K nonn,easy

%O 0,1

%A _L. Edson Jeffery_, Apr 20 2011