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 A189317 Expansion of 5*(1-6*x+x^2)/(1-10*x+5*x^2) 4
 5, 20, 180, 1700, 16100, 152500, 1444500, 13682500, 129602500, 1227612500, 11628112500, 110143062500, 1043290062500, 9882185312500, 93605402812500, 886643101562500, 8398404001562500, 79550824507812500, 753516225070312500, 7137408128164062500 (list; graph; refs; listen; history; text; internal format)
 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) (02, 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 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

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Last modified June 15 20:50 EDT 2019. Contains 324145 sequences. (Running on oeis4.)