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%I #21 Sep 08 2022 08:45:42
%S 392499,2281249,5732499,10746249,17322499,25461249,35162499,46426249,
%T 59252499,73641249,89592499,107106249,126182499,146821249,169022499,
%U 192786249,218112499,245001249,273452499,303466249,335042499,368181249
%N 781250n^2 - 455000n + 66249.
%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 a(n)^2-A157621(n)*A157622(n)^2=1.
%H Vincenzo Librandi, <a href="/A157623/b157623.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(1)=392499, a(2)=2281249, a(3)=5732499, a(n)=3*a(n-1)-3*a(n-2)+ a(n-3) [From Harvey P. Dale, Jul 11 2011]
%F G.f.: (392499+1103752*x+66249*x^2)/(1-x)^3 [From Harvey P. Dale, Jul 11 2011]
%e For n=1, a(1)=392499; n=2, a(2)=2281249; n=3, a(3)=5732499.
%t Table[781250n^2-455000n+66249,{n,25}] (* or *) LinearRecurrence[{3,-3,1},{392499,2281249,5732499},25] (* or *) CoefficientList[Series[ (-392499- 1103752 x-66249 x^2)/(x-1)^3,{x,0,25}],x] (* _Harvey P. Dale_, Jul 11 2011 *)
%o (Magma) I:=[392499, 2281249, 5732499]; [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)=781250*n^2-455000*n+66249
%Y Cf. A157621, A157622.
%K nonn,easy
%O 1,1
%A _Vincenzo Librandi_, Mar 03 2009
%E Minor corrections by _M. F. Hasler_, Oct 08 2014
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