A189317 Expansion of 5*(1-6*x+x^2)/(1-10*x+5*x^2)

5, 20, 180, 1700, 16100, 152500, 1444500, 13682500, 129602500, 1227612500, 11628112500, 110143062500, 1043290062500, 9882185312500, 93605402812500, 886643101562500, 8398404001562500, 79550824507812500, 753516225070312500, 7137408128164062500

Offset

0,1

Comments

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

A=A_(10,3)=

(0 0 0 1 0)

(0 0 1 0 1)

(0 1 0 2 0)

(1 0 2 0 1)

(0 2 0 2 0).

Then a(n)=Trace(A^(2*n)). (End)

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).

Links

Table of n, a(n) for n=0..19.

L. E. Jeffery, Unit-primitive matrices.

Index entries for linear recurrences with constant coefficients, signature (10, -5).

Formula

G.f.: 5*(1-6*x+x^2)/(1-10*x+5*x^2).

a(n)=10*a(n-1)-5*a(n-2), n>2, a(0)=5, a(1)=20, a(2)=180.

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

a(n)=2*((5-2*Sqrt(5))^n+(5+2*Sqrt(5))^n), for n>0, with a(0)=5.

Mathematica

CoefficientList[Series[5*(1-6x+x^2)/(1-10x+5x^2), {x, 0, 30}], x] (* or *) Join[ {5}, LinearRecurrence[{10, -5}, {20, 180}, 30]] (* Harvey P. Dale, Apr 02 2013 *)

Prog

(PARI) Vec(5*(1-6*x+x^2)/(1-10*x+5*x^2)+O(x^99)) \\ Charles R Greathouse IV, Sep 25 2012

Crossrefs

A189315, A189316, A189318.

Sequence in context: A205338 A197857 A197741 * A203902 A000877 A203113

Adjacent sequences: A189314 A189315 A189316 * A189318 A189319 A189320

Keyword

nonn,easy

Author

L. Edson Jeffery, Apr 20 2011

Status

approved