login
Values x of the solutions (x,y) of the Diophantine equation 5*(X-Y)^4 - 16*X*Y = 0 with X >= Y.
2

%I #19 Feb 15 2020 10:52:26

%S 0,40,11664,3733880,1201904928,387002605000,124613510434992,

%T 40125161088048920,12920177216118344256,4160256937698274701160,

%U 1339589813748515664595920,431343759769849048394285240

%N Values x of the solutions (x,y) of the Diophantine equation 5*(X-Y)^4 - 16*X*Y = 0 with X >= Y.

%C Corresponding Y values: b(n) = c(n)*(-1 + d(n)) (see Formula section for definitions of c(n) and d(n)), which gives 0, 32, 11520, 3731296, 1201858560, ...

%H G. C. Greubel, <a href="/A123381/b123381.txt">Table of n, a(n) for n = 0..395</a>

%F a(n) = c(n)*(1+d(n)) with c(0) = 0, c(1) = 4 and c(n) = 18*c(n-1) - c(n-2), d(0) = 1, d(1) = 9 and d(n) = 18*d(n-1) - d(n-2).

%F From _Max Alekseyev_, Nov 13 2009: (Start)

%F For n >= 4, a(n) = 340*a(n-1) - 5798*a(n-2) + 340*a(n-3) - a(n-4).

%F O.g.f.: 8*x*(5*x^2 - 242*x + 5)/((x^2 - 18*x + 1)*(x^2 - 322*x + 1)). (End)

%t CoefficientList[Series[8*x*(5*x^2 - 242*x + 5)/(x^2 - 18*x + 1)/(x^2 - 322*x + 1), {x, 0, 50}], x] (* _G. C. Greubel_, Oct 13 2017 *)

%o (PARI) x='x+O('x^50); concat([0], Vec(8*x*(5*x^2 -242*x +5)/((x^2 -18*x +1)*(x^2 -322*x +1)))) \\ _G. C. Greubel_, Oct 13 2017

%Y Equals 4*A123377. - _Michel Marcus_, Oct 14 2017

%K nonn

%O 0,2

%A _Mohamed Bouhamida_, Oct 13 2006

%E More terms from _Max Alekseyev_, Nov 13 2009