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A189317
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Expansion of 5*(1-6*x+x^2)/(1-10*x+5*x^2)
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4
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5, 20, 180, 1700, 16100, 152500, 1444500, 13682500, 129602500, 1227612500, 11628112500, 110143062500, 1043290062500, 9882185312500, 93605402812500, 886643101562500, 8398404001562500, 79550824507812500, 753516225070312500, 7137408128164062500
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OFFSET
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0,1
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COMMENTS
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(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).
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LINKS
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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).
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FORMULA
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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.
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MATHEMATICA
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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 *)
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PROG
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(PARI) Vec(5*(1-6*x+x^2)/(1-10*x+5*x^2)+O(x^99)) \\ Charles R Greathouse IV, Sep 25 2012
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CROSSREFS
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A189315, A189316, A189318.
Sequence in context: A205338 A197857 A197741 * A203902 A000877 A203113
Adjacent sequences: A189314 A189315 A189316 * A189318 A189319 A189320
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KEYWORD
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nonn,easy
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AUTHOR
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L. Edson Jeffery, Apr 20 2011
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STATUS
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approved
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