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%I #34 Sep 08 2022 08:45:42
%S 439,1760,3963,7048,11015,15864,21595,28208,35703,44080,53339,63480,
%T 74503,86408,99195,112864,127415,142848,159163,176360,194439,213400,
%U 233243,253968,275575,298064,321435,345688,370823,396840,423739,451520
%N a(n) = 441*n^2 - 2*n.
%C The identity (441*n - 1)^2 - (441*n^2 - 2*n)*21^2 = 1 can be written as A158319(n)^2 - a(n)*21^2 = 1 (see Barbeau's paper in link). Also, the identity (388962*n^2 - 1764*n + 1)^2 - (441*n^2 - 2*n)*(18522*n - 42)^2 = 1 can be written as A157739(n)^2 - a(n)*A157738(n)^2 = 1. - _Vincenzo Librandi_, Jan 25 2012
%C This last formula is the case s=21 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 05 2012
%H Vincenzo Librandi, <a href="/A157737/b157737.txt">Table of n, a(n) for n = 1..10000</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 (about the first identity in Comments section, row 15 in the first table at p. 85, case d(t) = t*(21^2*t-2)).
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F G.f.: x*(-439 - 443*x)/(x-1)^3.
%F a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
%t LinearRecurrence[{3,-3,1},{439,1760,3963},50]
%o (Magma) I:=[439, 1760, 3963]; [n le 3 select I[n] else 3*Self(n-1)-3*Self(n-2)+1*Self(n-3): n in [1..50]];
%o (PARI) a(n)=441*n^2-2*n \\ _Charles R Greathouse IV_, Dec 28 2011
%Y Cf. A158319, A157738, A157739.
%K nonn,easy
%O 1,1
%A _Vincenzo Librandi_, Mar 05 2009
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