A200216 Danilov's sequence of x satisfying 0 < |x^3-y^2| < sqrt(x) with integer (x,y).

93844, 322001299796379844, 1114592308630995805123571151844, 3858108676488182444301031186675778188809844, 13354661111806898918013326915229994453818137920195953844

Offset

1,1

Comments

For y values see A200217.

For x^3-y^2 values see A200218.

The values (a(n)+1)/5 are perfect squares: for sqrt((a(n)+1)/5) see A200335. - Jasinski

This sequence is an infinite subset of A078933. - Artur Jasinski, Nov 27 2011

References

For references see A078933 and A179386.

Links

Table of n, a(n) for n=1..5.

L. V. Danilov, Letter to the Editor, Mat. Zametki, 1984, Volume 36, Issue 3, Pages 457-458.

L. V. Danilov, Letter to the editor, Math. Notes 36 (3) (1984) 726.

R. D'Mello, Marshall Hall's Conjecture and Gaps Between Integer Points on Mordell Elliptic Curves, arXiv preprint arXiv:1410.0078 [math.NT], 2014.

Formula

Conjecture: a(n) = 3461450947505*a(n-1) + 6440022564931157490*a(n-2) - 6440022564931157490*a(n-3) - 3461450947505*a(n-4) + a(n-5). - R. J. Mathar, Nov 15 2011

Conjecture: g.f. 4092*(1-z)/(5*(1+1860498*z+z^2)) - 7/(10*(z-1)) + 930249*(1-z)/(10*(1-3461452808002*z+z^2)). - R. J. Mathar, Nov 15 2011

3125*A200218(n)^2 + 6750*A200218(n) + 729 = 2916*a(n). - Artur Jasinski, Nov 15 2011

125*y^2 *(54 + 50*x^3 - 25*y^2)=(9 - 6*x + 5*x^2)*(-9 + 15*x + 25*x^2)^2. - Artur Jasinski, Nov 16 2011

7/10 - (6/5)*(-1)^n * (1/2)*(f^(15*(2n-1))-(1/f)^(15(2n-1)))+(1/20)*(f^(30(2n-1))+(1/f)^(30(2n-1)))), where f is golden ratio constant = (1+sqrt(5)/2). - Artur Jasinski, Nov 17 2011

7/10+(3/5)*L(15*(2 n - 1))*(-1)^(n+1)+(1/20)*L(30*(2 n - 1)) where L(k) is the k-th Lucas number: A000204(n) or A000032(n+1). - Artur Jasinski, Nov 18 2011

Example

93844^3 -(round(sqrt(93844^3))^2 < sqrt(93844).

Mathematica

aa = {}; uu = 682 + 61*Sqrt[125]; Do[vv = Expand[uu^(2*n - 1)]; tt = ((-1)^n vv[[1]] + 57)/125; xx = (5^5*tt^2 - 3000*tt + 719); yy = Round[N[Sqrt[xx^3], 1000]]; dd = xx^3 - yy^2; AppendTo[aa, xx], {n, 1, 6}]; aa
(* second program follows the generator formula *)
data = Table[(7/10 - (6/5)*(-1)^n*(1/2)*(f^(15*(2 n - 1)) - (1/f)^(15 (2 n - 1))) + (1/20)*(f^(30 (2 n - 1)) + (1/f)^(30 (2 n - 1)))) /. f -> GoldenRatio, {n, 1, 6}]; data // FunctionExpand // ExpandAll // Simplify (* Bob Hanlon (hanlonr(AT)cox.net) *)
(* third program uses the Lucas numbers formula *)
Table[7/10 + (-1)^(n + 1) 3/5 LucasL[15*(2 n - 1)] +
  1/20 LucasL[30*(2 n - 1)] , {n, 1, 10}] (* Artur Jasinski, Nov 18 2011 *)

Prog

(PARI) u = quadunit(20)^5
for(i=1, 6, v = u^(2*i-1); t = ((-1)^i * real(v) + 57) / 125; print(5^5*t^2 - 3000*t + 719) ) \\ Noam D. Elkies

Crossrefs

Cf. A078933, A179107, A179108, A179109, A179387, A179388, A199496.

Sequence in context: A203784 A206683 A234215 * A126708 A209514 A029754

Adjacent sequences: A200213 A200214 A200215 * A200217 A200218 A200219

Keyword

nonn

Author

Artur Jasinski, Nov 14 2011

Status

approved