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a(n) = 80000*n^2 - 39200*n + 4801.
3

%I #20 Sep 08 2022 08:45:42

%S 45601,246401,607201,1128001,1808801,2649601,3650401,4811201,6132001,

%T 7612801,9253601,11054401,13015201,15136001,17416801,19857601,

%U 22458401,25219201,28140001,31220801,34461601,37862401,41423201,45144001

%N a(n) = 80000*n^2 - 39200*n + 4801.

%C The identity (80000*n^2 -39200*n +4801)^2 - (100*n^2 -49*n +6)*(8000*n -1960)^2 = 1 can be written as a(n)^2 - A157651(n)*A157652(n)^2 = 1.

%H Vincenzo Librandi, <a href="/A157653/b157653.txt">Table of n, a(n) for n = 1..10000</a>

%H Vincenzo Librandi, <a href="http://mathforum.org/kb/message.jspa?messageID=5773864&amp;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).

%F G.f.: x*(45601 + 109598*x + 4801*x^2)/(1-x)^3.

%F E.g.f.: (4801 + 40800*x + 80000*x^2)*exp(x) - 4801. - _G. C. Greubel_, Nov 17 2018

%t LinearRecurrence[{3,-3,1},{45601,246401,607201},40]

%o (Magma) I:=[45601, 246401, 607201]; [n le 3 select I[n] else 3*Self(n-1)-3*Self(n-2)+1*Self(n-3): n in [1..40]];

%o (PARI) a(n) = 80000*n^2 - 39200*n + 4801.

%o (Sage) [80000*n^2-39200*n+4801 for n in (1..40)] # _G. C. Greubel_, Nov 17 2018

%o (GAP) List([1..40], n -> 80000*n^2-39200*n+4801); # _G. C. Greubel_, Nov 17 2018

%Y Cf. A157651, A157652.

%K nonn,easy

%O 1,1

%A _Vincenzo Librandi_, Mar 03 2009