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%I #22 Sep 08 2022 08:45:42
%S 5967,25650,45333,65016,84699,104382,124065,143748,163431,183114,
%T 202797,222480,242163,261846,281529,301212,320895,340578,360261,
%U 379944,399627,419310,438993,458676,478359,498042,517725,537408,557091,576774,596457
%N a(n) = 19683*n - 13716.
%C The identity (531441*n^2 - 740664*n + 258065)^2 - (729*n^2 - 1016*n + 354)*(19683*n - 13716)^2 = 1 can be written as A157667(n)^2 - A157665(n)*a(n)^2 = 1.
%H Harvey P. Dale, <a href="/A157666/b157666.txt">Table of n, a(n) for n = 1..1000</a>
%H Vincenzo Librandi, <a href="http://mathforum.org/kb/message.jspa?messageID=5773864&tstart=0">X^2-AY^2=1</a>
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (2,-1).
%F From _Harvey P. Dale_, Nov 03 2011: (Start)
%F G.f.: 27*x*(508*x+221)/(x-1)^2.
%F a(n) = 2*a(n-1) - a(n-2); a(1)=5967, a(2)=25650. (End)
%F E.g.f.: 27*(508 - (508 - 729*x)*exp(x)). - _G. C. Greubel_, Nov 17 2018
%e a(1) = 19683*1 - 13716 = 5967;
%e a(2) = 19683*2 - 13716 = 25650.
%t 19683Range[40]-13716 (* or *) LinearRecurrence[{2,-1},{5967,25650},40] (* _Harvey P. Dale_, Nov 03 2011 *)
%o (Magma) I:=[5967, 25650]; [n le 2 select I[n] else 2*Self(n-1)-Self(n-2): n in [1..40]];
%o (PARI) a(n) = 19683*n - 13716.
%o (Sage) [19683*n-13716 for n in (1..40)] # _G. C. Greubel_, Nov 17 2018
%o (GAP) List([1..40], n -> 19683*n-13716); # _G. C. Greubel_, Nov 17 2018
%Y Cf. A157665, A157667.
%K nonn,easy
%O 1,1
%A _Vincenzo Librandi_, Mar 04 2009
%E Example edited by _Jon E. Schoenfield_, Nov 17 2018
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