|
%I #21 Apr 18 2023 09:45:57
%S 730,1459,2188,2917,3646,4375,5104,5833,6562,7291,8020,8749,9478,
%T 10207,10936,11665,12394,13123,13852,14581,15310,16039,16768,17497,
%U 18226,18955,19684,20413,21142,21871,22600,23329,24058,24787,25516,26245
%N a(n) = 729n + 1.
%C The identity (729*n+1)^2-(729*n^2+2*n)*(27)^2 = 1 can be written as a(n)^2-A158396(n)*(27)^2 = 1.
%H Vincenzo Librandi, <a href="/A158397/b158397.txt">Table of n, a(n) for n = 1..10000</a>
%H Vincenzo Librandi, <a href="http://mathforum.org/kb/message.jspa?messageID=5785989&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_02">Index entries for linear recurrences with constant coefficients</a>, signature (2,-1).
%F G.f.: x*(730-x)/(1-x)^2.
%F a(n) = 2*a(n-1)-a(n-2).
%t LinearRecurrence[{2,-1},{730,1459},50]
%t 729*Range[40]+1 (* _Harvey P. Dale_, Apr 08 2014 *)
%o (Magma) I:=[730, 1459]; [n le 2 select I[n] else 2*Self(n-1)-Self(n-2): n in [1..50]];
%o (PARI) a(n) = 729*n + 1.
%Y Cf. A158396.
%K nonn,easy
%O 1,1
%A _Vincenzo Librandi_, Mar 18 2009
|
