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A158231 a(n) = 256*n + 1.

%I A158231 #38 Jul 09 2025 14:26:23
%S A158231 257,513,769,1025,1281,1537,1793,2049,2305,2561,2817,3073,3329,3585,
%T A158231 3841,4097,4353,4609,4865,5121,5377,5633,5889,6145,6401,6657,6913,
%U A158231 7169,7425,7681,7937,8193,8449,8705,8961,9217,9473,9729,9985,10241,10497
%N A158231 a(n) = 256*n + 1.
%C A158231 The identity (256*n + 1)^2 - (256*n^2 + 2*n)*16^2 = 1 can be written as a(n)^2 - A158230(n)*16^2 = 1.
%C A158231 Also the identity (512*n + 1)^2 - (256*n^2 + n)*32^2 = 1 can be written as A076338(n)^2 - (n*a(n))*32^2 = 1. - _Vincenzo Librandi_, Feb 23 2012
%H A158231 Vincenzo Librandi, <a href="/A158231/b158231.txt">Table of n, a(n) for n = 1..10000</a>
%H A158231 Vincenzo Librandi, <a href="http://mathforum.org/kb/message.jspa?messageID=5785989&amp;tstart=0">X^2-AY^2=1</a>
%H A158231 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 (first identity in the comment section: row 15 in the initial table at p. 85, case d(t) = t*(16^2*t+2)).
%H A158231 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (2,-1).
%F A158231 a(n) = 2*a(n-1) - a(n-2); a(1)=257, a(2)=513. - _Harvey P. Dale_, Nov 21 2011
%F A158231 G.f.: x*(257-x)/(x-1)^2. - _Harvey P. Dale_, Nov 21 2011
%p A158231 A158231:=n->256*n + 1; seq(A158231(n), n=1..50); # _Wesley Ivan Hurt_, Jan 24 2014
%t A158231 256Range[50]+1 (* or *) LinearRecurrence[{2,-1},{257,513},50] (* _Harvey P. Dale_, Nov 21 2011 *)
%o A158231 (Magma) I:=[257, 513]; [n le 2 select I[n] else 2*Self(n-1)-Self(n-2): n in [1..50]];
%o A158231 (PARI) a(n) = 256*n + 1
%Y A158231 Cf. A076338, A158230.
%K A158231 nonn,easy
%O A158231 1,1
%A A158231 _Vincenzo Librandi_, Mar 14 2009