%I #20 Sep 08 2022 08:45:42
%S 12799,51841,117127,208657,326431,470449,640711,837217,1059967,
%T 1308961,1584199,1885681,2213407,2567377,2947591,3354049,3786751,
%U 4245697,4730887,5242321,5779999,6343921,6934087,7550497,8193151,8862049
%N a(n) = 13122*n^2 - 324*n + 1.
%C The identity (13122*n^2 - 324*n + 1)^2 - (81*n^2 - 2*n)*(1458*n - 18)^2 = 1 can be written as a(n)^2 - A157507(n)* A157508(n)^2 = 1. - _Vincenzo Librandi_, Jan 26 2012
%C This 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_, Jan 26 2011
%H Vincenzo Librandi, <a href="/A157509/b157509.txt">Table of n, a(n) for n = 1..10000</a>
%H Vincenzo Librandi, <a href="http://mathforum.org/kb/message.jspa?messageID=5771301&tstart=0">X^2-AY^2=1</a>
%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 26 2012
%F G.f.: x*(-12799 - 13444*x - x^2)/(x-1)^3. - _Vincenzo Librandi_, Jan 26 2012
%t LinearRecurrence[{3,-3,1},{12799,51841,117127},40] (* _Vincenzo Librandi_, Jan 26 2012 *)
%t Table[13122n^2-324n+1,{n,30}] (* _Harvey P. Dale_, Jun 30 2022 *)
%o (Magma) I:=[12799, 51841, 117127]; [n le 3 select I[n] else 3*Self(n-1)-3*Self(n-2)+1*Self(n-3): n in [1..40]]; // _Vincenzo Librandi_, Jan 26 2012
%o (PARI) for(n=1, 22, print1(13122n^2 - 324n + 1", ")); \\ _Vincenzo Librandi_, Jan 26 2012
%Y Cf. A157507, A157508.
%K nonn,easy
%O 1,1
%A _Vincenzo Librandi_, Mar 02 2009