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%I #27 Oct 05 2022 22:58:05
%S 314,1825,4586,8597,13858,20369,28130,37141,47402,58913,71674,85685,
%T 100946,117457,135218,154229,174490,196001,218762,242773,268034,
%U 294545,322306,351317,381578,413089,445850,479861,515122,551633,589394,628405
%N a(n) = 625n^2 - 364n + 53.
%C The identity (781250*n^2-455000*n+66249)^2-(625*n^2-364*n+53)*(31250*n-9100)^2=1 can be written as A157623(n)^2-a(n)*A157622(n)^2=1.
%C The continued fraction expansion of sqrt(a(n)) is [25n-8; {1, 2, 1, 1, 2, 1, 50n-16}]. - _Magus K. Chu_, Oct 05 2022
%H Vincenzo Librandi, <a href="/A157621/b157621.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> [Dead link]
%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*(-314-883*x-53*x^2)/(x-1)^3.
%p A157621:= n-> 625*n^2 - 364*n + 53: seq(A157621(n), n=1..40); # _Wesley Ivan Hurt_, Feb 15 2014
%t LinearRecurrence[{3,-3,1},{314,1825,4586},40]
%o (Magma) I:=[314, 1825, 4586]; [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) = 625*n^2 - 364*n + 53;
%Y Cf. A157622, A157623.
%K nonn,easy
%O 1,1
%A _Vincenzo Librandi_, Mar 03 2009
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