A158382 - OEIS

%I #23 Sep 08 2022 08:45:43

%S 627,2504,5631,10008,15635,22512,30639,40016,50643,62520,75647,90024,

%T 105651,122528,140655,160032,180659,202536,225663,250040,275667,

%U 302544,330671,360048,390675,422552,455679,490056,525683,562560,600687,640064

%N a(n) = 625*n^2 + 2*n.

%C The identity (625*n+1)^2 - (625*n^2+2*n)*(25)^2 = 1 can be written as A158383(n)^2 - a(n)*(25)^2 = 1.

%H Vincenzo Librandi, <a href="/A158382/b158382.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*(25^2*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*(-627-623*x)/(x-1)^3. [corrected by _Georg Fischer_, May 11 2019]

%t LinearRecurrence[{3,-3,1},{627,2504,5631},50]

%o (Magma) I:=[627, 2504, 5631]; [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) = 625*n^2 + 2*n.

%Y Cf. A158383.

%K nonn,easy

%O 1,1

%A _Vincenzo Librandi_, Mar 17 2009