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%I #19 Sep 08 2022 08:45:43
%S 624,1249,1874,2499,3124,3749,4374,4999,5624,6249,6874,7499,8124,8749,
%T 9374,9999,10624,11249,11874,12499,13124,13749,14374,14999,15624,
%U 16249,16874,17499,18124,18749,19374,19999,20624,21249,21874,22499,23124
%N 625n - 1.
%C The identity (625*n-1)^2-(625*n^2-2*n)*(25)^2=1 can be written as a(n)^2-A158373(n)*(25)^2=1.
%H Vincenzo Librandi, <a href="/A158374/b158374.txt">Table of n, a(n) for n = 1..10000</a>
%H Vincenzo Librandi, <a href="http://mathforum.org/kb/message.jspa?messageID=5785989&tstart=0">X^2-AY^2=1</a>
%H E. J. Barbeau, <a href="http://www.math.toronto.edu/barbeau/home.html">Polynomial Excursions</a>, Chapter 10: <a href="http://www.math.toronto.edu/barbeau/hxpol10.pdf">Diophantine equations</a> (2010), pages 84-85 (row 15 in the first table at p. 85, case d(t) = t*(25^2*t-2)).
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (2,-1).
%F a(n) = 2*a(n-1)-a(n-2).
%F G.f.: x*(624+x)/(1-x)^2.
%t LinearRecurrence[{2,-1},{624,1249},50]
%t 625*Range[50]-1 (* _Harvey P. Dale_, May 26 2018 *)
%o (Magma) I:=[624, 1249]; [n le 2 select I[n] else 2*Self(n-1)-Self(n-2): n in [1..50]];
%o (PARI) a(n) = 625*n - 1.
%Y Cf. A158373.
%K nonn,easy
%O 1,1
%A _Vincenzo Librandi_, Mar 17 2009