%I #22 Sep 08 2022 08:45:42
%S 896845951,4685313601,11438583751,21156656401,33839531551,49487209201,
%T 68099689351,89676972001,114219057151,141725944801,172197634951,
%U 205634127601,242035422751,281401520401,323732420551,369028123201,417288628351
%N a(n) = 1482401250*n^2 - 658736100*n + 73180801.
%C The identity (1482401250*n^2 - 658736100*n + 73180801)^2 - (27225*n^2 - 12098*n + 1344)*(8984250*n - 1996170)^2 = 1 can be written as a(n)^2 - A157796(n)*A157797(n)^2 = 1.
%C This is the case s=165 and r=6049 of the identity (2*(s^2*n-r)^2-1)^2 - (((s^2*n-r)^2-1)/s^2)*(2*s*(s^2*n-r))^2 = 1, where ((s^2*n-r)^2-1)/s^2 is an integer if r^2 == 1 (mod s^2). Therefore, for s=165, nonnegative r values are: 1, 1574, 6049, 7624, 19601, 21176, 25651, ... - _Bruno Berselli_, Apr 24 2018
%H Vincenzo Librandi, <a href="/A157798/b157798.txt">Table of n, a(n) for n = 1..10000</a>
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F G.f.: x*(896845951 + 1994775748*x + 73180801*x^2)/(1 - x)^3.
%F a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
%t LinearRecurrence[{3, -3, 1}, {896845951, 4685313601, 11438583751}, 30]
%o (Magma) I:=[896845951, 4685313601, 11438583751]; [n le 3 select I[n] else 3*Self(n-1)-3*Self(n-2)+Self(n-3): n in [1..30]];
%o (PARI) a(n) = 1482401250*n^2 - 658736100*n + 73180801;
%Y Cf. A157796, A157797.
%K nonn,easy
%O 1,1
%A _Vincenzo Librandi_, Mar 07 2009