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 A189315 Expansion of 5*(1-3*x+x^2)/(1-5*x+5*x^2). 8
 5, 10, 30, 100, 350, 1250, 4500, 16250, 58750, 212500, 768750, 2781250, 10062500, 36406250, 131718750, 476562500, 1724218750, 6238281250, 22570312500, 81660156250, 295449218750, 1068945312500, 3867480468750, 13992675781250, 50625976562500 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,1 COMMENTS Let A be the unit-primitive matrix (see [Jeffery]) A=A_(10,1)= (0 1 0 0 0) (1 0 1 0 0) (0 1 0 1 0) (0 0 1 0 1) (0 0 0 2 0). Then a(n) = Trace(A^(2*n)). 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)=10, a(2)=30. a(n) = Sum_{k=1..5) (w_k)^(2*n), w_k=2*cos((2*k-1)*Pi/10). a(n) = 2^(1-n)*((5-Sqrt(5))^n+(5+Sqrt(5))^n), for n>0, with a(0)=5. a(n) = 5*A147748(n). MATHEMATICA CoefficientList[Series[5(1-3x+x^2)/(1-5x+5x^2), {x, 0, 40}], x] (* or *) Join[{5}, LinearRecurrence[{5, -5}, {10, 30}, 40]]  (* Harvey P. Dale, Apr 25 2011 *) PROG (PARI) Vec(5*(1-3*x+x^2)/(1-5*x+5*x^2)+O(x^99)) \\ Charles R Greathouse IV, Sep 24 2012 (MAGMA) I:=[5, 10, 30]; [n le 3 select I[n] else 5*Self(n-1)-5*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Dec 09 2015 CROSSREFS Cf. A147748, A189316, A189317, A189318. Cf. A025192, A081567, A127672, A265185. Sequence in context: A048010 A002571 A077916 * A056422 A032296 A052648 Adjacent sequences:  A189312 A189313 A189314 * A189316 A189317 A189318 KEYWORD nonn,easy AUTHOR L. Edson Jeffery, Apr 20 2011 STATUS approved

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Last modified July 21 19:25 EDT 2019. Contains 325199 sequences. (Running on oeis4.)