A123116 Values y of solutions (x, y) to the Diophantine equation (x-y)^4 - 8*x*y = 0 with x >= y.

0, 4, 192, 6860, 235008, 7994836, 271656000, 9228697244, 313506312192, 10649999100580, 361786539945408, 12290092806887276, 417501371504448000, 14182756553557856884, 481796221538133532992, 16366888776259793834300, 555992422174307055403008

Offset

0,2

Comments

Corresponding x-values (A123057) are x(n) = c(n)*(1 + d(n)) with c(n) and d(n) defined in formula section.

The pair (x,y) = (A001542(n), a(n)) satisfies the equation 2*x^4 - 2*x*y - y^2 = 0. - Alexander Samokrutov, Sep 04 2015

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) with c(0)=0, c(1)=2 and c(n) = 6*c(n-1) - c(n-2) d(0)=1, d(1)=3 and 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.: 4*x*(1 +8*x +x^2)/((1 -34*x +x^2)*(1 -6*x +x^2)). - Colin Barker, Oct 25 2012

a(n) = A123057(n) - 2*A001542(n). - Alexander Samokrutov, Sep 05 2015

a(n) = (1/2)*(A000129(4*n) - 2*A000129(2*n)) = (1/2)*A000129(2*n)*(A002203(2*n) - 2) = 2*A123115(n). - G. C. Greubel, Jul 21 2021

Mathematica

LinearRecurrence[{40, -206, 40, -1}, {0, 4, 192, 6860}, 40] (* Vincenzo Librandi, Sep 22 2015 *)
Table[(Fibonacci[4*n, 2] - 2*Fibonacci[2*n, 2])/2, {n, 0, 30}] (* G. C. Greubel, Jul 21 2021 *)

Prog

(PARI) concat(0, Vec(4*x*(1+8*x+x^2)/((1-34*x+x^2)*(1-6*x+x^2)) + O(x^20))) \\ Michel Marcus, Sep 05 2015
(Magma) I:=[0, 4, 192, 6860]; [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..20]]; // Vincenzo Librandi, Sep 22 2015
(Sage) [(1/2)*(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, A123057, A123115.

Sequence in context: A208184 A355613 A172809 * A163839 A012015 A012102

Adjacent sequences: A123113 A123114 A123115 * A123117 A123118 A123119

Keyword

nonn,easy

Author

Mohamed Bouhamida, Sep 28 2006

Extensions

More terms from Max Alekseyev, Nov 13 2009

Edited by Michel Marcus, Sep 05 2015

Status

approved