A181679 - OEIS

%I #45 Sep 15 2024 01:44:39

%S 123,488,1095,1944,3035,4368,5943,7760,9819,12120,14663,17448,20475,

%T 23744,27255,31008,35003,39240,43719,48440,53403,58608,64055,69744,

%U 75675,81848,88263,94920,101819,108960,116343,123968,131835,139944,148295,156888,165723,174800,184119,193680

%N a(n) = 121*n^2 + 2*n.

%C The identity (121*n + 1)^2 - (121*n^2 + 2*n)*11^2 = 1 can be written as A158131(n)^2 - a(n)*11^2 = 1 (see Barbeau's paper in link).

%C Also, the identity (29282*n^2 + 484*n + 1)^2 - (121*n^2 + 2*n)*(2662*n + 22)^2 = 1 can be written as A157614(n)^2 - a(n)*A157613(n)^2 = 1. - _Vincenzo Librandi_, Feb 21 2012

%C This last formula is the case s=11 of the identity (2*s^4*n^2 + 4*s^2*n + 1)^2 - (s^2*n^2 + 2*n)*(2*s^3*n + 2*s)^2 = 1. - _Bruno Berselli_, Feb 21 2012

%H Vincenzo Librandi, <a href="/A181679/b181679.txt">Table of n, a(n) for n = 1..10000</a>

%H Vincenzo Librandi, <a href="http://mathforum.org/kb/message.jspa?messageID=5771301&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*(11^2*t+2)).

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).

%F a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).

%F G.f.: x*(-119*x - 123)/(x-1)^3.

%t LinearRecurrence[{3,-3,1},{123,488,1095},50]

%o (Magma) [ n*(121*n+2): n in [1..40] ];

%o (PARI) a(n) = n*(121*n+2)

%Y Cf. A157613, A157614, A158131.

%K nonn,easy

%O 1,1

%A _Vincenzo Librandi_, Nov 18 2010, Nov 19 2010