A189318 Expansion of 5*(1-2*x)/(1-3*x-2*x^2+4*x^3).

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

Offset: 0

Author

L. Edson Jeffery, Apr 20 2011

Keywords

• nonn
• easy

Comments

(Start) Let A be the unit-primitive 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 unit-primitive matrix A_(N,r) (0

Links

• L. E. Jeffery, Unit-primitive matrices.
• Index entries for linear recurrences with constant coefficients, signature (3, 2, -4).

Crossrefs

Cf. A052899.

A189315, A189316, A189317.

Programs

Mathematica

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 *)

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

Formula

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

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.

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

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

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

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