login
Values y of solutions (x, y) to the Diophantine equation (x-y)^4 - 8*x*y = 0 with x >= y.
2

%I #43 Sep 08 2022 08:45:28

%S 0,4,192,6860,235008,7994836,271656000,9228697244,313506312192,

%T 10649999100580,361786539945408,12290092806887276,417501371504448000,

%U 14182756553557856884,481796221538133532992,16366888776259793834300,555992422174307055403008

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

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

%C 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

%H G. C. Greubel, <a href="/A123116/b123116.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) 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).

%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.: 4*x*(1 +8*x +x^2)/((1 -34*x +x^2)*(1 -6*x +x^2)). - _Colin Barker_, Oct 25 2012

%F a(n) = A123057(n) - 2*A001542(n). - _Alexander Samokrutov_, Sep 05 2015

%F 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

%t LinearRecurrence[{40, -206, 40, -1}, {0, 4, 192, 6860}, 40] (* _Vincenzo Librandi_, Sep 22 2015 *)

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

%o (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

%o (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

%o (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

%Y Cf. A000129, A002203, A123057, A123115.

%K nonn,easy

%O 0,2

%A _Mohamed Bouhamida_, Sep 28 2006

%E More terms from _Max Alekseyev_, Nov 13 2009

%E Edited by _Michel Marcus_, Sep 05 2015