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%I #31 Feb 26 2022 18:05:08
%S 0,180,233640,303264540,393637139280,510940703520900,
%T 663200639532988920,860833919173116097260,1117361763886065161254560,
%U 1450334708690193406192321620,1882533334518107155172472208200,2443526817869794397220462733921980
%N Numbers k such that 13*k^2 + 1 is a square.
%C Limit_{n->infinity} a(n)/a(n-1) = 649 + 180*sqrt(13).
%C This sequence gives the values of y in solutions of the Diophantine equation x^2 - 13*y^2 = 1. The corresponding x values are in A114047. - _Vincenzo Librandi_, Aug 08 2010, edited by _Jon E. Schoenfield_, May 04 2014
%H Colin Barker, <a href="/A075871/b075871.txt">Table of n, a(n) for n = 1..322</a>
%H Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (1298,-1).
%F a(n) = 1298*a(n-1) - a(n-2), n>1. - _Michael Somos_, Oct 30 2002
%F a(n) = ((649 + 180*sqrt(13))^n - (649 - 180*sqrt(13))^n) / (2*sqrt(13)).
%F From _Mohamed Bouhamida_, Sep 20 2006: (Start)
%F a(n) = 1297*(a(n-1) + a(n-2)) - a(n-3).
%F a(n) = 1299*(a(n-1) - a(n-2)) + a(n-3). (End)
%F G.f.: 180*x^2/(1-1298*x+x^2). - _Philippe Deléham_, Nov 18 2008
%t LinearRecurrence[{1298, -1}, {0, 180}, 50] (* _Vincenzo Librandi_, Jun 14 2015 *)
%o (PARI) concat(0, Vec(180*x^2/(1-1298*x+x^2) + O(x^20))) \\ _Colin Barker_, Jun 13 2015
%o (Magma) I:=[0,180]; [n le 2 select I[n] else 1298*Self(n-1)-Self(n-2): n in [1..20]]; // _Vincenzo Librandi_, Jun 14 2015
%Y Cf. A114047.
%Y Cf. A202156.
%K nonn,easy
%O 1,2
%A _Gregory V. Richardson_, Oct 16 2002
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