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A123057 Values x of solutions (x, y) to the Diophantine equation (x-y)^4 - 8*x*y = 0 with x >= y. 2
0, 8, 216, 7000, 235824, 7999592, 271683720, 9228858808, 313507253856, 10650004589000, 361786571934264, 12290092993331992, 417501372591127440, 14182756559891488808, 481796221575048645096, 16366888776474950875000, 555992422175561082535104 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Corresponding y-values (A123116) are y(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)*(1 + d(n)) 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.: 8*x*(1 -13*x +x^2)/((1-34*x+x^2)*(1-6*x+x^2)). - Colin Barker, Oct 24 2012

a(n) = A123116(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*A123056(n).  - G. C. Greubel, Jul 20 2021

MATHEMATICA

CoefficientList[Series[8*x*(1-13*x+x^2)/((1-34*x+x^2)*(1-6*x+x^2)), {x, 0, 30}], x] (* Vincenzo Librandi, Sep 04 2015 *)

Table[(Fibonacci[4*n, 2] + 2*Fibonacci[2*n, 2])/2, {n, 0, 30}] (* G. C. Greubel, Jul 20 2021 *)

PROG

(PARI) concat(0, Vec(8*x*(1-13*x+x^2)/((1-34*x+x^2)*(1-6*x+x^2)) + O(x^20))) \\ Michel Marcus, Sep 05 2015

(MAGMA) R<x>:=PowerSeriesRing(Integers(), 30); [0] cat Coefficients(R!( 8*x*(1 -13*x +x^2)/((1-34*x+x^2)*(1-6*x+x^2)) )); // G. C. Greubel, Jul 20 2021

(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 20 2021

CROSSREFS

Cf. A000129, A001542, A002203, A123056, A123116.

Sequence in context: A288323 A264056 A271400 * A009072 A002897 A024289

Adjacent sequences:  A123054 A123055 A123056 * A123058 A123059 A123060

KEYWORD

nonn,easy

AUTHOR

Mohamed Bouhamida, Sep 26 2006

EXTENSIONS

More terms from Max Alekseyev, Nov 13 2009

a(16) from Vincenzo Librandi, Sep 04 2015

Edited by Michel Marcus, Sep 05 2015

STATUS

approved

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Last modified September 21 12:54 EDT 2021. Contains 347598 sequences. (Running on oeis4.)