A158123 - OEIS

%I #23 Sep 08 2022 08:45:42

%S 82,163,244,325,406,487,568,649,730,811,892,973,1054,1135,1216,1297,

%T 1378,1459,1540,1621,1702,1783,1864,1945,2026,2107,2188,2269,2350,

%U 2431,2512,2593,2674,2755,2836,2917,2998,3079,3160,3241,3322,3403,3484,3565

%N a(n) = 81*n + 1.

%C The identity (81*n + 1)^2 - (81*n^2 + 2*n)*9^2 = 1 can be written as a(n)^2 - A177099(n)*9^2 = 1. - _Vincenzo Librandi_, Feb 03 2012

%H Vincenzo Librandi, <a href="/A158123/b158123.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 15 in the first table at p. 85, case d(t) = t*(9^2*t+2)).

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (2,-1).

%F G.f.: x*(82-x)/(1-x)^2. - _Vincenzo Librandi_, Feb 03 2012

%F a(n) = 2*a(n-1) - a(n-2). - _Vincenzo Librandi_, Feb 03 2012

%t LinearRecurrence[{2, -1}, {82, 163}, 50] (* _Vincenzo Librandi_, Feb 03 2012 *)

%o (Magma) I:=[82, 163]; [n le 2 select I[n] else 2*Self(n-1)-Self(n-2): n in [1..50]]; // _Vincenzo Librandi_, Feb 03 2012

%o (PARI) for(n=1, 50, print1(81*n + 1", ")); \\ _Vincenzo Librandi_, Feb 03 2012

%Y Cf. A177099.

%K nonn,easy

%O 1,1

%A _Vincenzo Librandi_, Mar 13 2009