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324n^2 - 2n.
2

%I #21 Sep 08 2022 08:45:42

%S 322,1292,2910,5176,8090,11652,15862,20720,26226,32380,39182,46632,

%T 54730,63476,72870,82912,93602,104940,116926,129560,142842,156772,

%U 171350,186576,202450,218972,236142,253960,272426,291540,311302,331712

%N 324n^2 - 2n.

%C The identity (324*n-1)^2-(324*n^2-2*n)*(18)^2=1 can be written as A158306(n)^2-a(n)*(18)^2=1.

%H Vincenzo Librandi, <a href="/A158305/b158305.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*(18^2*t-2)).

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).

%F Contribution from Harvey P. Dale, Jul 14 2011: (Start)

%F G.f.: -2*x*(163*x+161)/(x-1)^3.

%F a(1)=322, a(2)=1292, a(3)=2910, a(n)=3*a(n-1)-3*a(n-2)+a(n-3). (End)

%t Table[324n^2-2n,{n,40}] (* or *) LinearRecurrence[{3,-3,1},{322,1292,2910},40] (* _Harvey P. Dale_, Jul 14 2011 *)

%o (Magma) I:=[322, 1292, 2910]; [n le 3 select I[n] else 3*Self(n-1)-3*Self(n-2)+1*Self(n-3): n in [1..50]];

%o (PARI) a(n) = 324*n^2 - 2*n.

%Y Cf. A158306.

%K nonn,easy

%O 1,1

%A _Vincenzo Librandi_, Mar 16 2009