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%I #24 Sep 08 2022 08:45:42
%S 71,143,215,287,359,431,503,575,647,719,791,863,935,1007,1079,1151,
%T 1223,1295,1367,1439,1511,1583,1655,1727,1799,1871,1943,2015,2087,
%U 2159,2231,2303,2375,2447,2519,2591,2663,2735,2807,2879,2951,3023,3095,3167,3239
%N a(n) = 72*n - 1.
%C The identity (72*n - 1)^2 - (36*n^2 - n)*12^2 = 1 can be written as a(n)^2 - A157286(n)*12^2 = 1. - _Vincenzo Librandi_, Jan 28 2012
%H Vincenzo Librandi, <a href="/A157921/b157921.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 14 in the first table at p. 85, case d(t) = t*(6^2*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). - _Vincenzo Librandi_, Jan 28 2012
%F G.f.: x*(x+71)/(x-1)^2. - _Vincenzo Librandi_, Jan 28 2012
%t LinearRecurrence[{2,-1},{71,143},50] (* _Vincenzo Librandi_, Jan 28 2012 *)
%t 72*Range[50]-1 (* _Harvey P. Dale_, Sep 06 2019 *)
%o (Magma) I:=[71, 143]; [n le 2 select I[n] else 2*Self(n-1)-Self(n-2): n in [1..50]]; // _Vincenzo Librandi_, Jan 28 2012
%o (PARI) for(n=1, 40, print1(72*n - 1", ")); \\ _Vincenzo Librandi_, Jan 28 2012
%Y Cf. A157286.
%K nonn,easy
%O 1,1
%A _Vincenzo Librandi_, Mar 09 2009
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