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 A157912 a(n) = 64*n^2 + 16. 2
 80, 272, 592, 1040, 1616, 2320, 3152, 4112, 5200, 6416, 7760, 9232, 10832, 12560, 14416, 16400, 18512, 20752, 23120, 25616, 28240, 30992, 33872, 36880, 40016, 43280, 46672, 50192, 53840, 57616, 61520, 65552, 69712, 74000, 78416, 82960, 87632, 92432, 97360, 102416 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS The identity (8*n^2+1)^2 - (64*n^2+16)*n^2 = 1 can be written as A081585(n)^2 - a(n)*n^2 = 1. - Vincenzo Librandi, Feb 09 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 09 2012: (Start) G.f.: x*(80+32*x+16*x^2)/(1-x)^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) = (coth(Pi/2)*Pi/2 - 1)/32. Sum_{n>=1} (-1)^(n+1)/a(n) = (1 - cosech(Pi/2)*Pi/2)/32. (End) MAPLE A157912:=n->64*n^2 + 16: seq(A157912(n), n=1..50); # Wesley Ivan Hurt, Oct 27 2014 MATHEMATICA 64Range[50]^2+16 (* Harvey P. Dale, Mar 24 2011 *) LinearRecurrence[{3, -3, 1}, {80, 272, 592}, 40] (* Vincenzo Librandi, Feb 09 2012 *) PROG (Magma) I:=[80, 272, 592]; [n le 3 select I[n] else 3*Self(n-1)-3*Self(n-2)+Self(n-3): n in [1..50]]; // Vincenzo Librandi, Feb 09 2012 (PARI) for(n=1, 40, print1(64*n^2 + 16", ")); \\ Vincenzo Librandi, Feb 09 2012 CROSSREFS Cf. A081585. Sequence in context: A235091 A306434 A234890 * A234883 A194646 A057441 Adjacent sequences: A157909 A157910 A157911 * A157913 A157914 A157915 KEYWORD nonn,easy AUTHOR Vincenzo Librandi, Mar 09 2009 STATUS approved

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Last modified February 28 13:24 EST 2024. Contains 370399 sequences. (Running on oeis4.)