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%I #27 Sep 08 2022 08:45:42
%S 197,393,589,785,981,1177,1373,1569,1765,1961,2157,2353,2549,2745,
%T 2941,3137,3333,3529,3725,3921,4117,4313,4509,4705,4901,5097,5293,
%U 5489,5685,5881,6077,6273,6469,6665,6861,7057,7253,7449,7645,7841,8037,8233,8429
%N a(n) = 196*n + 1.
%C The identity (196*n + 1)^2 - (196*n^2 + 2*n)*14^2 = 1 can be written as a(n)^2 - A158222(n)*14^2 = 1.
%C Also, the identity (392*n + 1)^2 - (196*n^2 + n)*28^2 = 1 can be written as A158002(n)^2 - (n*a(n))*28^2 = 1. - _Vincenzo Librandi_, Feb 23 2012
%H Vincenzo Librandi, <a href="/A158223/b158223.txt">Table of n, a(n) for n = 1..10000</a>
%H Vincenzo Librandi, <a href="http://mathforum.org/kb/message.jspa?messageID=5785989&tstart=0">X^2-AY^2=1</a>
%H E. J. Barbeau, <a href="http://www.math.toronto.edu/barbeau/home.html">Polynomial Excursions</a>, Chapter 10: <a href="http://www.math.toronto.edu/barbeau/hxpol10.pdf">Diophantine equations</a> (2010), pages 84-85 (first identity in the comment section: row 15 in the initial table at p. 85, case d(t) = t*(14^2*t+2); second identity: row 14, case d(t) = t*(14^2*t+1)).
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (2,-1).
%F a(n) = 2*a(n-1) - a(n-2).
%F G.f.: x*(197-x)/(1-x)^2.
%t LinearRecurrence[{2,-1},{197,393},50]
%t 196 Range[50]+1 (* _Harvey P. Dale_, Jul 23 2021 *)
%o (Magma) I:=[197, 393]; [n le 2 select I[n] else 2*Self(n-1)-Self(n-2): n in [1..50]];
%o (PARI) a(n) = 196*n + 1.
%Y Cf. A158222, A158002.
%K nonn,easy
%O 1,1
%A _Vincenzo Librandi_, Mar 14 2009
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