A158249 - OEIS

%I #28 Feb 07 2025 14:25:24

%S 254,1020,2298,4088,6390,9204,12530,16368,20718,25580,30954,36840,

%T 43238,50148,57570,65504,73950,82908,92378,102360,112854,123860,

%U 135378,147408,159950,173004,186570,200648,215238,230340,245954,262080,278718,295868

%N a(n) = 256*n^2 - 2*n.

%C The identity (256*n - 1)^2 - (256*n^2 - 2*n)*16^2 = 1 can be written as A158250(n)^2 - a(n)*16^2 = 1.

%H Vincenzo Librandi, <a href="/A158249/b158249.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 (row 15 in the first table at p. 85, case d(t) = t*(16^2*t-2)).

%H Vincenzo Librandi, <a href="http://mathforum.org/kb/message.jspa?messageID=5785989&amp;tstart=0">X^2-AY^2=1</a>

%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*(254 + 258*x)/(1-x)^3.

%F E.g.f.: 2*x*(127 + 128*x)*exp(x). - _G. C. Greubel_, Apr 24 2022

%t LinearRecurrence[{3,-3,1},{254,1020,2298},50]

%o (Magma) I:=[254, 1020, 2298]; [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) = 256*n^2-2*n

%o (SageMath) [2*n*(128*n-1) for n in (1..50)] # _G. C. Greubel_, Apr 24 2022

%Y Cf. A158250.

%K nonn,easy,changed

%O 1,1

%A _Vincenzo Librandi_, Mar 15 2009