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a(n) = 200*n - 1.
2

%I #17 Dec 27 2023 22:32:27

%S 199,399,599,799,999,1199,1399,1599,1799,1999,2199,2399,2599,2799,

%T 2999,3199,3399,3599,3799,3999,4199,4399,4599,4799,4999,5199,5399,

%U 5599,5799,5999,6199,6399,6599,6799,6999,7199,7399,7599,7799,7999,8199,8399,8599

%N a(n) = 200*n - 1.

%C The identity (200*n-1)^2-(100*n^2-n)*(20)^2=1 can be written as a(n)^2-A157659(n)*(20)^2=1.

%H Vincenzo Librandi, <a href="/A157955/b157955.txt">Table of n, a(n) for n = 1..10000</a>

%H Vincenzo Librandi, <a href="http://mathforum.org/kb/message.jspa?messageID=5785989&amp;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 14 in the first table at p. 85, case d(t) = t*(10^2*t-1)).

%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*(199+x)/(1-x)^2.

%t LinearRecurrence[{2,-1},{199,399},50]

%o (Magma) I:=[199, 399]; [n le 2 select I[n] else 2*Self(n-1)-Self(n-2): n in [1..50]];

%o (PARI) a(n) = 200*n - 1.

%Y Cf. A157659.

%K nonn,easy

%O 1,1

%A _Vincenzo Librandi_, Mar 10 2009