A200218 The differences x^3 - y^2 of Danilov's subsequence of good Hall's examples A078933.

-297, 548147655, -1019827620252441, 1897387247823873407415, -3530085179800800999132960777, 6567716416847133270037051381858983, -12219223258107727669457593220846745613305, 22733840433256343397153666138928891468676446359

Offset

1,1

Comments

For x values see A200216.

For y values see A200217.

All terms in this sequence are of the form: 3^3 * 11(2^3 * 31 * 61^2 * k + 922807).

Links

Table of n, a(n) for n=1..8.

Formula

3125 * a(n)^2 + 6750 * a(n) + 729 = 2916 * A200216(n).

a(n) = (A200216(n))^3 - (A200217(n))^2.

Conjecture: a(n) = -1860497 * a(n-1) + 1860497 * a(n-2) + a(n-3) with g.f. 297 * z * (1 + 14882 * z + z^2) / ( (z-1)*(z^2 + 1860498 * z+1) ). - R. J. Mathar, Nov 15 2011

Hyperelliptic curve (157464*y)^2 = (729 + 594*d + 125*d^2) (-729 + 13500*d + 15625*d^2)^2 is singular (has two cusps) and for this reason Danilov's sequence has infinitely many integer solutions. - Artur Jasinski, Nov 16 2011

(27/125) * (-5 + (-1)^n * ((-1)^(n+1) * 6 + L[15(2n - 1)]) where L(k) is the k-th Lucas number: A000204(n) or A000032(n+1). - Artur Jasinski, Nov 18 2011

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, dd], {n, 1, 10}]; aa
(* Recurrence generator of R. J. Mathar *)
dd = {-297, 548147655, -1019827620252441}; a0 = dd[[1]]; a1 = dd[[2]]; a2 = dd[[3]]; Do[a = a0 + 1860497 * a1 - 1860497 * a2; a0 = a1; a1 = a2; a2 = a; AppendTo[aa, a], {n, 1, 10}]; aa
(* Third one after Lucas numbers formula *)
Table[27/125 (-5 + (-1)^n ((-1)^(n + 1) 6 + LucasL[15 (-1 + 2 n)])), {n, 10}] (* Artur Jasinski, Nov 18 2011*)

Crossrefs

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

Sequence in context: A200855 A200582 A200581 * A284156 A234665 A234659

Adjacent sequences: A200215 A200216 A200217 * A200219 A200220 A200221

Keyword

sign

Author

Artur Jasinski, Nov 14 2011

Status

approved