A177099 - OEIS

%I #46 Sep 08 2022 08:45:53

%S 83,328,735,1304,2035,2928,3983,5200,6579,8120,9823,11688,13715,15904,

%T 18255,20768,23443,26280,29279,32440,35763,39248,42895,46704,50675,

%U 54808,59103,63560,68179,72960,77903,83008,88275,93704,99295,105048,110963,117040,123279,129680,136243,142968,149855,156904,164115

%N a(n) = 81*n^2 + 2*n.

%C The identity (81*n + 1)^2 - (81*n^2 + 2*n)*9^2 = 1 can be written as A158123(n)^2 - a(n)*9^2 = 1 (see Barbeau's paper in link). - _Vincenzo Librandi_, Feb 03 2012

%C Also, the identity (13122*n^2 + 324*n + 1)^2 - (81*n^2 + 2*n)*(1458*n + 18)^2 = 1 can be written as A157506(n)^2 - a(n)*A157505(n)^2 = 1. - _Vincenzo Librandi_, Feb 04 2012

%C This last formula is the case s=9 of the identity (2*s^4*n^2 + 4*s^2*n + 1)^2 - (s^2*n^2 + 2*n)*(2*s^3*n + 2*s)^2 = 1. - _Bruno Berselli_, Feb 04 2011

%H Vincenzo Librandi, <a href="/A177099/b177099.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*(9^2*t+2)).

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

%F G.f.: x*(-79*x - 83)/(x - 1)^3. - _Harvey P. Dale_, Mar 23 2011

%F a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - _Vincenzo Librandi_, Feb 03 2012

%t Table[81n^2+2n,{n,50}]  (* _Harvey P. Dale_, Mar 23 2011 *)

%t LinearRecurrence[{3, -3, 1}, {83, 328, 735}, 40] (* _Vincenzo Librandi_, Feb 03 2012 *)

%o (Magma) [ 81*n^2+2*n: n in [1..50] ];

%o (PARI) for(n=1, 50, print1(81*n^2+2*n", ")); \\ _Vincenzo Librandi_, Feb 03 2012

%Y Cf. A157505, A157506, A158123.

%K nonn,easy

%O 1,1

%A _Vincenzo Librandi_, Nov 18 2010

%E G.f. adapted to the offset by _Bruno Berselli_, Apr 01 2011