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 A157910 a(n) = 18*n^2 - 1. 3
 17, 71, 161, 287, 449, 647, 881, 1151, 1457, 1799, 2177, 2591, 3041, 3527, 4049, 4607, 5201, 5831, 6497, 7199, 7937, 8711, 9521, 10367, 11249, 12167, 13121, 14111, 15137, 16199, 17297, 18431, 19601, 20807, 22049, 23327, 24641, 25991, 27377, 28799, 30257, 31751 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS The identity (18*n^2 - 1)^2 - (81*n^2 - 9)*(2*n)^2 = 1 can be written as a(n)^2 - A157909(n)*A005843(n)^2 = 1. - Vincenzo Librandi, Feb 08 2012 LINKS Vincenzo Librandi, Table of n, a(n) for n = 1..10000 Vincenzo Librandi, X^2-AY^2=1, Math Forum, 2007. [Wayback Machine link] Index entries for linear recurrences with constant coefficients, signature (3,-3,1). FORMULA From Vincenzo Librandi, Feb 08 2012: (Start) G.f.: x*(-17 - 20*x + x^2)/(x - 1)^3. a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). (End) From Amiram Eldar, Mar 07 2023: (Start) Sum_{n>=1} 1/a(n) = (1 - cot(Pi/(3*sqrt(2)))*Pi/(3*sqrt(2)))/2. Sum_{n>=1} (-1)^(n+1)/a(n) = (cosec(Pi/(3*sqrt(2)))*Pi/(3*sqrt(2)) - 1)/2. (End) MATHEMATICA 18Range[40]^2-1 (* Harvey P. Dale, Mar 24 2011 *) LinearRecurrence[{3, -3, 1}, {17, 71, 161}, 50] (* Vincenzo Librandi, Feb 08 2012 *) PROG (Magma) I:=[17, 71, 161]; [n le 3 select I[n] else 3*Self(n-1)-3*Self(n-2)+1*Self(n-3): n in [1..40]]; // Vincenzo Librandi, Feb 08 2012 (PARI) for(n=1, 40, print1(18*n^2 - 1", ")); \\ Vincenzo Librandi, Feb 08 2012 CROSSREFS Cf. A157909, A005843. Sequence in context: A044010 A106921 A105414 * A141959 A347334 A290243 Adjacent sequences: A157907 A157908 A157909 * A157911 A157912 A157913 KEYWORD nonn,easy AUTHOR Vincenzo Librandi, Mar 09 2009 STATUS approved

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Last modified July 24 13:23 EDT 2024. Contains 374584 sequences. (Running on oeis4.)