A189236 Expansion of (5-8*x-15*x^2+4*x^3+4*x^4)/(1-2*x-5*x^2+2*x^3+4*x^4+x^5).

5, 2, 14, 32, 114, 347, 1142, 3649, 11826, 38111, 123139, 397443, 1283406, 4143479, 13378435, 43194542, 139463234, 450284986, 1453839839, 4694021537, 15155624819, 48933074467, 157990585613, 510105367936, 1646980994190, 5317619734147

Offset: 0

Author

L. Edson Jeffery, Apr 18 2011

Keywords

• nonn
• easy

Comments

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

U=U_(11,3)=

(0 0 0 1 0)

(0 0 1 0 1)

(0 1 0 1 1)

(1 0 1 1 1)

(0 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

Links

• L. E. Jeffery, Unit-primitive matrices
• Index entries for linear recurrences with constant coefficients, signature (2, 5, -2, -4, -1).

Crossrefs

Cf. A062883, A189234, A189235, A189237.

Programs

Mathematica

CoefficientList[Series[ (5-8x-15x^2+4x^3+4x^4)/ (1-2x-5x^2+2x^3+4x^4+x^5), {x,0,29}],x]  (* Harvey P. Dale, Apr 19 2011 *)
LinearRecurrence[{2, 5, -2, -4, -1}, {5, 2, 14, 32, 114}, 30] (* T. D. Noe, Apr 19 2011 *)

PARI

Vec((5-8*x-15*x^2+4*x^3+4*x^4)/(1-2*x-5*x^2+2*x^3+4*x^4+x^5)+O(x^99)) \\ Charles R Greathouse IV, Sep 25 2012

Formula

G.f.: (5-8*x-15*x^2+4*x^3+4*x^4)/(1-2*x-5*x^2+2*x^3+4*x^4+x^5).

a(n)=2*a(n-1)+5*a(n-2)-2*a(n-3)-4*a(n-4)-a(n-5), {a(m)}={5,2,14,32,114}, m=0..4.

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

Series expansion of g.f. at x=infinity gives -A062883 and all but the first term of -A189235.

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