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%I #20 Aug 26 2023 14:54:28
%S 145,289,433,577,721,865,1009,1153,1297,1441,1585,1729,1873,2017,2161,
%T 2305,2449,2593,2737,2881,3025,3169,3313,3457,3601,3745,3889,4033,
%U 4177,4321,4465,4609,4753,4897,5041,5185,5329,5473,5617,5761,5905,6049,6193
%N a(n) = 144*n + 1.
%C The identity (144*n+1)^2-(144*n^2+2*n)*(12)^2=1 can be written as a(n)^2-A158132(n)*(12)^2=1.
%H Vincenzo Librandi, <a href="/A158133/b158133.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 (row 15 in the first table at p. 85, case d(t) = t*(12^2*t+2)).
%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*(145-x)/(1-x)^2.
%t LinearRecurrence[{2,-1},{145,289},50]
%o (Magma) I:=[145, 289]; [n le 2 select I[n] else 2*Self(n-1)-Self(n-2): n in [1..50]];
%o (PARI) a(n) = 144n + 1.
%Y Cf. A158132.
%K nonn,easy
%O 1,1
%A _Vincenzo Librandi_, Mar 13 2009