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Values of Y satisfying the equation (X-Y)^4 - 2*X*Y = 0 with integer X >= Y >= 0.
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%I #21 Sep 08 2022 08:45:28

%S 0,2,96,3430,117504,3997418,135828000,4614348622,156753156096,

%T 5324999550290,180893269972704,6145046403443638,208750685752224000,

%U 7091378276778928442,240898110769066766496,8183444388129896917150

%N Values of Y satisfying the equation (X-Y)^4 - 2*X*Y = 0 with integer X >= Y >= 0.

%C Values of X are given by c(n)*(d(n) + 1) and listed in A123056.

%H G. C. Greubel, <a href="/A123115/b123115.txt">Table of n, a(n) for n = 0..400</a>

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (40,-206,40,-1).

%F a(n) = c(n)*(d(n) - 1), where c(0)=0, c(1)=1, c(n) = 6*c(n-1) - c(n-2) and d(0)=1, d(1)=3, d(n) = 6*d(n-1) - d(n-2).

%F For n>=4, a(n) = 40*a(n-1) - 206*a(n-2) + 40*a(n-3) - a(n-4). - _Max Alekseyev_, Nov 13 2009

%F G.f.: 2*x*(1 +8*x +x^2)/((1 -34*x +x^2)*(1 -6*x +x^2)). - _Colin Barker_, Nov 04 2012

%F a(n) = (1/4)*(A000129(4*n) - 2*A000129(2*n)) = (1/4)*A000129(2*n)*(A002203(2*n) - 2). - _G. C. Greubel_, Jul 21 2021

%t LinearRecurrence[{40,-206,40,-1},{0,2,96,3430},20] (* _Harvey P. Dale_, May 28 2015 *)

%t Table[(Fibonacci[4*n, 2] - 2*Fibonacci[2*n, 2])/4, {n, 0, 30}] (* _G. C. Greubel_, Jul 21 2021 *)

%o (Magma) I:=[0, 2, 96, 3430]; [n le 4 select I[n] else 40*Self(n-1) -206*Self(n-2) +40*Self(n-3) -Self(n-4): n in [1..31]]; // _G. C. Greubel_, Jul 21 2021

%o (Sage) [(1/4)*(lucas_number1(4*n, 2, -1) - 2*lucas_number1(2*n, 2, -1)) for n in (0..30)] # _G. C. Greubel_, Jul 21 2021

%Y Cf. A000129, A002203, A123056.

%K nonn,easy

%O 0,2

%A _Mohamed Bouhamida_, Sep 28 2006

%E More terms from _Max Alekseyev_, Nov 13 2009