A189237 Expansion of (5-12*x-9*x^2+8*x^3+x^4)/(1-3*x-3*x^2+4*x^3+x^4-x^5)

5, 3, 15, 42, 155, 533, 1884, 6604, 23219, 81555, 286555, 1006734, 3537032, 12426742, 43659386, 153390077, 538911123, 1893376346, 6652069455, 23370962220, 82110068595, 288480349402, 1013528712002, 3560868017067, 12510529683224

Offset

0,1

Comments

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

U=U_(11,4)=

(0 0 0 0 1)

(0 0 0 1 1)

(0 0 1 1 1)

(0 1 1 1 1)

(1 1 1 1 1).

Then a(n)=Trace(U^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 U_(N,r) (0<r<floor(N/2)) and for which the closed-form expression for a(n) is derived from the eigenvalues of U_(N,r).

Links

Table of n, a(n) for n=0..24.

L. E. Jeffery, Unit-primitive matrices

Index entries for linear recurrences with constant coefficients, signature (3, 3, -4, -1, 1).

Formula

G.f.: (5-12*x-9*x^2+8*x^3+x^4)/(1-3*x-3*x^2+4*x^3+x^4-x^5).

a(n)=3*a(n-1)+3*a(n-2)-4*a(n-3)-a(n-4)+a(n-5), {a(m)}={5,3,15,42,155}, m=0..4.

a(n)=Sum_{k=1..5} ((x_k)^4-3*(x_k)^2+1)^n; x_k=2*(-1)^(k-1)*cos(k*Pi/11).

Series expansion of g.f. at x=infinity gives -A189234(n+1).

Mathematica

CoefficientList[Series[(5-12x-9x^2+8x^3+x^4)/(1-3x-3x^2+4x^3+x^4-x^5), {x, 0, 30}], x] (* or *) LinearRecurrence[{3, 3, -4, -1, 1}, {5, 3, 15, 42, 155}, 30] (* Harvey P. Dale, Oct 01 2011 *)

Prog

(PARI) Vec((5-12*x-9*x^2+8*x^3+x^4)/(1-3*x-3*x^2+4*x^3+x^4-x^5)+O(x^99)) \\ Charles R Greathouse IV, Sep 26 2012

Crossrefs

A189234, A189235, A189236.

Sequence in context: A073677 A223535 A211943 * A301494 A053371 A199005

Adjacent sequences: A189234 A189235 A189236 * A189238 A189239 A189240

Keyword

nonn,easy

Author

L. Edson Jeffery, Apr 18 2011

Status

approved