

A189318


Expansion of 5*(12*x)/(13*x2*x^2+4*x^3)


5



5, 5, 25, 65, 225, 705, 2305, 7425, 24065, 77825, 251905, 815105, 2637825, 8536065, 27623425, 89391105, 289275905, 936116225, 3029336065, 9803137025, 31723618305, 102659784705, 332214042625, 1075067224065, 3478990618625, 11258250133505
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OFFSET

0,1


COMMENTS

(Start) Let A be the unitprimitive matrix (see [Jeffery])
A=A_(10,4)=
(0 0 0 0 1)
(0 0 0 2 0)
(0 0 2 0 1)
(0 2 0 2 0)
(2 0 2 0 1).
Then a(n)=Trace(A^n). (End)
Evidently one of a class of accelerator sequences for Catalan's constant based on traces of successive powers of a unitprimitive matrix A_(N,r) (0<r<floor(N/2)) and for which the closedform expression for a(n) is derived from the eigenvalues of A_(N,r).


LINKS

Table of n, a(n) for n=0..25.
L. E. Jeffery, Unitprimitive matrices.
Index entries for linear recurrences with constant coefficients, signature (3, 2, 4).


FORMULA

G.f.: 5*(12*x)/(13*x2*x^2+4*x^3).
a(n)=3*a(n1)+2*a(n2)4*a(n3), n>3, a(0)=5, a(1)=5, a(2)=25, a(3)=65.
a(n)=Sum_{k=1..5} ((w_k)^43*(w_k)^2+1)^n, w_k=2*cos((2*k1)*Pi/10).
a(n)=1+2*(1Sqrt(5))^n+2*(1+Sqrt(5))^n.
a(n)=5*A052899(n).


MATHEMATICA

CoefficientList[Series[5(12x)/(13x2x^2+4x^3), {x, 0, 30}], x] (* or *) LinearRecurrence[{3, 2, 4}, {5, 5, 25}, 30] (* Harvey P. Dale, Jun 02 2014 *)


PROG

(PARI) Vec(5*(12*x)/(13*x2*x^2+4*x^3)+O(x^99)) \\ Charles R Greathouse IV, Sep 25 2012


CROSSREFS

Cf. A052899.
A189315, A189316, A189317.
Sequence in context: A257607 A093643 A223263 * A257615 A257624 A176160
Adjacent sequences: A189315 A189316 A189317 * A189319 A189320 A189321


KEYWORD

nonn,easy


AUTHOR

L. Edson Jeffery, Apr 20 2011


STATUS

approved



