%I #21 Sep 08 2022 08:45:42
%S 79201,318401,717601,1276801,1996001,2875201,3914401,5113601,6472801,
%T 7992001,9671201,11510401,13509601,15668801,17988001,20467201,
%U 23106401,25905601,28864801,31984001,35263201,38702401,42301601,46060801
%N a(n) = 80000*n^2 - 800*n + 1.
%C The identity (80000*n^2 - 800*n + 1)^2 - (100*n^2 - n)*(8000*n - 40)^2 = 1 can be written as a(n)^2 - A157659(n)* A157660(n)^2 = 1. This is the case s=10 of the identity (8*n^2*s^4 - 8*n*s^2 + 1)^2 - (n^2*s^2 - n)*(8*n*s^3 - 4*s)^2 = 1. - _Vincenzo Librandi_, Jan 28 2012
%H Vincenzo Librandi, <a href="/A157661/b157661.txt">Table of n, a(n) for n = 1..10000</a>
%H Vincenzo Librandi, <a href="http://mathforum.org/kb/message.jspa?messageID=5773864&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 28 2012
%F G.f.: x*(-79201 - 80798*x - x^2)/(x-1)^3. - _Vincenzo Librandi_, Jan 28 2012
%F E.g.f.: (1 + 79200*x + 80000*x^2)*exp(x) - 1. _G. C. Greubel_, Nov 17 2018
%t LinearRecurrence[{3,-3,1},{79201,318401,717601},40] (* _Vincenzo Librandi_, Jan 28 2012 *)
%o (Magma) I:=[79201, 318401, 717601]; [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 28 2012
%o (PARI) for(n=1, 40, print1(80000*n^2 - 800*n + 1, ", ")); \\ _Vincenzo Librandi_, Jan 28 2012
%o (Sage) [80000*n^2-800*n+1 for n in (1..40)] # _G. C. Greubel_, Nov 17 2018
%o (GAP) List([1..40], n -> 80000*n^2-800*n+1); # _G. C. Greubel_, Nov 17 2018
%Y Cf. A157659, A157660.
%K nonn,easy
%O 1,1
%A _Vincenzo Librandi_, Mar 04 2009