A123115 Values of Y satisfying the equation (X-Y)^4 - 2*X*Y = 0 with integer X >= Y >= 0.

0, 2, 96, 3430, 117504, 3997418, 135828000, 4614348622, 156753156096, 5324999550290, 180893269972704, 6145046403443638, 208750685752224000, 7091378276778928442, 240898110769066766496, 8183444388129896917150

Offset

0,2

Comments

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

Links

G. C. Greubel, Table of n, a(n) for n = 0..400

Index entries for linear recurrences with constant coefficients, signature (40,-206,40,-1).

Formula

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).

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

G.f.: 2*x*(1 +8*x +x^2)/((1 -34*x +x^2)*(1 -6*x +x^2)). - Colin Barker, Nov 04 2012

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

Mathematica

LinearRecurrence[{40, -206, 40, -1}, {0, 2, 96, 3430}, 20] (* Harvey P. Dale, May 28 2015 *)
Table[(Fibonacci[4*n, 2] - 2*Fibonacci[2*n, 2])/4, {n, 0, 30}] (* G. C. Greubel, Jul 21 2021 *)

Prog

(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
(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

Crossrefs

Cf. A000129, A002203, A123056.

Sequence in context: A069121 A362406 A157065 * A119696 A264542 A064069

Adjacent sequences: A123112 A123113 A123114 * A123116 A123117 A123118

Keyword

nonn,easy

Author

Mohamed Bouhamida, Sep 28 2006

Extensions

More terms from Max Alekseyev, Nov 13 2009

Status

approved