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529n - 1.
2

%I #19 Sep 08 2022 08:45:43

%S 528,1057,1586,2115,2644,3173,3702,4231,4760,5289,5818,6347,6876,7405,

%T 7934,8463,8992,9521,10050,10579,11108,11637,12166,12695,13224,13753,

%U 14282,14811,15340,15869,16398,16927,17456,17985,18514,19043,19572

%N 529n - 1.

%C The identity (529*n-1)^2-(529*n^2-2*n)*(23)^2=1 can be written as a(n)^2-A158364(n)*(23)^2=1.

%H Vincenzo Librandi, <a href="/A158365/b158365.txt">Table of n, a(n) for n = 1..10000</a>

%H Vincenzo Librandi, <a href="http://mathforum.org/kb/message.jspa?messageID=5785989&amp;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*(23^2*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*(528+x)/(1-x)^2.

%t LinearRecurrence[{2,-1},{528,1057},50]

%t 529*Range[40]-1 (* _Harvey P. Dale_, Jul 08 2017 *)

%o (Magma) I:=[528, 1057]; [n le 2 select I[n] else 2*Self(n-1)-Self(n-2): n in [1..50]];

%o (PARI) a(n) = 529*n - 1.

%Y Cf. A158364.

%K nonn,easy

%O 1,1

%A _Vincenzo Librandi_, Mar 17 2009