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%I #18 Sep 08 2022 08:45:42
%S 80,322,726,1292,2020,2910,3962,5176,6552,8090,9790,11652,13676,15862,
%T 18210,20720,23392,26226,29222,32380,35700,39182,42826,46632,50600,
%U 54730,59022,63476,68092,72870,77810,82912,88176,93602,99190,104940
%N a(n) = 81n^2 - n.
%C The identity (162*n - 1)^2 - (81*n^2 - n)*18^2 = 1 can be written as A157954(n)^2 - a(n)*18^2 = 1. - _Vincenzo Librandi_, Jan 29 2012 [corrected by _Jon E. Schoenfield_, Aug 18 2018]
%H Vincenzo Librandi, <a href="/A157953/b157953.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*(9^2*t-1)).
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - _Vincenzo Librandi_, Jan 29 2012
%F G.f.: x*(-80-82*x)/(x-1)^3. - _Vincenzo Librandi_, Jan 29 2012
%t LinearRecurrence[{3,-3,1},{80,322,726},50] (* _Vincenzo Librandi_, Jan 29 2012 *)
%o (Magma) I:=[80, 322, 726]; [n le 3 select I[n] else 3*Self(n-1)-3*Self(n-2)+1*Self(n-3): n in [1..50]]; // _Vincenzo Librandi_, Jan 29 2012
%o (PARI) for(n=1, 40, print1(81*n^2 - n", ")); \\ _Vincenzo Librandi_, Jan 29 2012
%Y Cf. A157954.
%K nonn,easy
%O 1,1
%A _Vincenzo Librandi_, Mar 10 2009
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