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A158729 a(n) = 1156*n^2 - 34.

%I A158729 #25 Jan 16 2025 16:27:17
%S A158729 1122,4590,10370,18462,28866,41582,56610,73950,93602,115566,139842,
%T A158729 166430,195330,226542,260066,295902,334050,374510,417282,462366,
%U A158729 509762,559470,611490,665822,722466,781422,842690,906270,972162,1040366,1110882,1183710,1258850,1336302
%N A158729 a(n) = 1156*n^2 - 34.
%C A158729 The identity (68*n^2 - 1)^2 - (1156*n^2 - 34)*(2*n)^2 = 1 can be written as A158730(n)^2 - a(n)*A005843(n)^2 = 1.
%H A158729 Vincenzo Librandi, <a href="/A158729/b158729.txt">Table of n, a(n) for n = 1..10000</a>
%H A158729 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 A158729 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F A158729 G.f.: 34*x*(-33 - 36*x + x^2)/(x-1)^3.
%F A158729 a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
%F A158729 From _Amiram Eldar_, Mar 22 2023: (Start)
%F A158729 Sum_{n>=1} 1/a(n) = (1 - cot(Pi/sqrt(34))*Pi/sqrt(34))/68.
%F A158729 Sum_{n>=1} (-1)^(n+1)/a(n) = (cosec(Pi/sqrt(34))*Pi/sqrt(34) - 1)/68. (End)
%F A158729 From _Elmo R. Oliveira_, Jan 16 2025: (Start)
%F A158729 E.g.f.: 34*(exp(x)*(34*x^2 + 34*x - 1) + 1).
%F A158729 a(n) = 34*A158588(n). (End)
%t A158729 LinearRecurrence[{3, -3, 1}, {1122, 4590, 10370}, 50] (* _Vincenzo Librandi_, Feb 20 2012 *)
%o A158729 (Magma) I:=[1122, 4590, 10370]; [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 20 2012
%o A158729 (PARI) for(n=1, 40, print1(1156*n^2 - 34", ")); \\ _Vincenzo Librandi_, Feb 20 2012
%Y A158729 Cf. A005843, A158588, A158730.
%K A158729 nonn,easy
%O A158729 1,1
%A A158729 _Vincenzo Librandi_, Mar 25 2009
%E A158729 Comment rewritten and formula replaced by _R. J. Mathar_, Oct 22 2009