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A158589 a(n) = 324*n^2 - 18.

%I A158589 #29 Jan 16 2025 09:39:14
%S A158589 306,1278,2898,5166,8082,11646,15858,20718,26226,32382,39186,46638,
%T A158589 54738,63486,72882,82926,93618,104958,116946,129582,142866,156798,
%U A158589 171378,186606,202482,219006,236178,253998,272466,291582,311346,331758,352818,374526,396882,419886
%N A158589 a(n) = 324*n^2 - 18.
%C A158589 The identity (36*n^2 - 1)^2 - (324*n^2 - 18)*(2*n)^2 = 1 can be written as A136017(n)^2 - a(n)*A005843(n)^2 = 1.
%H A158589 Vincenzo Librandi, <a href="/A158589/b158589.txt">Table of n, a(n) for n = 1..10000</a>
%H A158589 Vincenzo Librandi, <a href="https://web.archive.org/web/20090309225914/http://mathforum.org/kb/message.jspa?messageID=5785989&amp;tstart=0">X^2-AY^2=1</a>, Math Forum, 2007. [Wayback Machine link]
%H A158589 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F A158589 From _Vincenzo Librandi_, Feb 16 2012: (Start)
%F A158589 G.f.: -18*x*(17 + 20*x - x^2)/(x-1)^3.
%F A158589 a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). (End)
%F A158589 From _Amiram Eldar_, Mar 14 2023: (Start)
%F A158589 Sum_{n>=1} 1/a(n) = (1 - cot(Pi/(3*sqrt(2)))*Pi/(3*sqrt(2)))/36.
%F A158589 Sum_{n>=1} (-1)^(n+1)/a(n) = (cosec(Pi/(3*sqrt(2)))*Pi/(3*sqrt(2)) - 1)/36. (End)
%F A158589 From _Elmo R. Oliveira_, Jan 16 2025: (Start)
%F A158589 E.g.f.: 18*(exp(x)*(18*x^2 + 18*x - 1) + 1).
%F A158589 a(n) = 18*A157910(n). (End)
%t A158589 LinearRecurrence[{3, -3, 1}, {306, 1278, 2898}, 40] (* _Vincenzo Librandi_, Feb 16 2012 *)
%t A158589 324*Range[40]^2-18 (* _Harvey P. Dale_, Jul 25 2019 *)
%o A158589 (Magma) I:=[306, 1278, 2898]; [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 16 2012
%o A158589 (PARI) for(n=1, 40, print1(324*n^2 - 18", ")); \\ _Vincenzo Librandi_, Feb 16 2012
%Y A158589 Cf. A005843, A136017, A157910.
%K A158589 nonn,easy
%O A158589 1,1
%A A158589 _Vincenzo Librandi_, Mar 22 2009
%E A158589 Comment rewritten by _R. J. Mathar_, Oct 28 2009