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%I #58 Jan 01 2024 23:50:27
%S 0,4,80,1444,25920,465124,8346320,149768644,2687489280,48225038404,
%T 865363202000,15528312597604,278644263554880,5000068431390244,
%U 89722587501469520,1610006506595061124,28890394531209630720,518417095055178291844,9302617316461999622480
%N a(0)=0, a(1)=4; for n > 1, a(n) = 18*a(n-1) - a(n-2) + 8.
%C The old definition given for this sequence was "Sequence allows us to find X values of the equation: X(X + 1) - 5*Y^2 = 0".
%C With this old definition, if X = a(n), then Y = A207832(n). Now, with u = 2X+1, this Diophantine equation becomes the Pell-Fermat equation u^2 - 20*Y^2 = 1, and then, u = A023039(n) and Y = A207832(n). - _Bernard Schott_, Jan 25 2023
%H Seiichi Manyama, <a href="/A132584/b132584.txt">Table of n, a(n) for n = 0..500</a>
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (19,-19,1).
%F a(n) = (A023039(n) - 1)/2. - _Max Alekseyev_, Nov 13 2009
%F G.f.: -4*x*(x+1)/((x-1)*(x^2-18*x+1)). - _Colin Barker_, Oct 24 2012
%F From _Amiram Eldar_, Jan 11 2022: (Start)
%F a(n) = 5*Fibonacci(3*n)^2/4 - 1 if n is odd and 5*Fibonacci(3*n)^2/4 if n is even.
%F A000217(a(n)) = A292443(n). (End)
%F a(n) = (Lucas(6*n)-2)/4. - _Jeffrey Shallit_, Jan 20 2023
%F a(n) = 4 * A049683(n). - _Alois P. Heinz_, Jan 20 2023
%t LinearRecurrence[{19, -19, 1}, {0, 4, 80}, 40] (* _Vincenzo Librandi_, Dec 24 2018 *)
%o (Magma) I:=[0,4,80]; [n le 3 select I[n] else 18*Self(n-1)-Self(n-2)+8: n in [1..30]]; // _Vincenzo Librandi_, Dec 24 2018
%Y Cf. A000032, A000045, A000217, A007654, A023039, A049683, A292443, A207832.
%K nonn,easy
%O 0,2
%A _Mohamed Bouhamida_, Nov 14 2007
%E More terms from _Max Alekseyev_, Nov 13 2009
%E New definition by _Antti Karttunen_, Oct 24 2012
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