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A158396 729n^2 + 2n. 2


%S 731,2920,6567,11672,18235,26256,35735,46672,59067,72920,88231,105000,

%T 123227,142912,164055,186656,210715,236232,263207,291640,321531,

%U 352880,385687,419952,455675,492856,531495,571592,613147,656160,700631,746560

%N 729n^2 + 2n.

%C The identity (729*n+1)^2-(729*n^2+2*n)*(27)^2=1 can be written as A158397(n)^2-a(n)*(27)^2=1.

%H Vincenzo Librandi, <a href="/A158396/b158396.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*(27^2*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*(731+727*x)/(1-x)^3.

%t LinearRecurrence[{3,-3,1},{731,2920,6567},50]

%o (MAGMA) I:=[731, 2920, 6567]; [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) = 729*n^2 + 2*n.

%Y Cf. A158397.

%K nonn,easy

%O 1,1

%A _Vincenzo Librandi_, Mar 18 2009

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Last modified January 21 16:22 EST 2020. Contains 331114 sequences. (Running on oeis4.)