A158271 - OEIS

%I #18 Nov 14 2024 08:53:16

%S 326,1300,2922,5192,8110,11676,15890,20752,26262,32420,39226,46680,

%T 54782,63532,72930,82976,93670,105012,117002,129640,142926,156860,

%U 171442,186672,202550,219076,236250,254072,272542,291660,311426,331840

%N 324n^2 + 2n.

%C The identity (324*n+1)^2-(324*n^2+2*n)*(18)^2=1 can be written as A158272(n)^2-a(n)*(18)^2=1.

%H Vincenzo Librandi, <a href="/A158271/b158271.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*(18^2*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*(-322*x-326)/(x-1)^3.

%t LinearRecurrence[{3,-3,1},{326,1300,2922},50]

%o (Magma) I:=[326, 1300, 2922]; [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) = 324*n^2+2*n

%Y Cf. A158272.

%K nonn,easy,changed

%O 1,1

%A _Vincenzo Librandi_, Mar 15 2009