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%I #52 Dec 19 2016 05:24:59
%S -297,548147655,-1019827620252441,1897387247823873407415,
%T -3530085179800800999132960777,6567716416847133270037051381858983,
%U -12219223258107727669457593220846745613305,22733840433256343397153666138928891468676446359
%N The differences x^3 - y^2 of Danilov's subsequence of good Hall's examples A078933.
%C For x values see A200216.
%C For y values see A200217.
%C All terms in this sequence are of the form: 3^3 * 11(2^3 * 31 * 61^2 * k + 922807).
%F 3125 * a(n)^2 + 6750 * a(n) + 729 = 2916 * A200216(n).
%F a(n) = (A200216(n))^3 - (A200217(n))^2.
%F 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
%F 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
%F (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
%t 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
%t (* Recurrence generator of _R. J. Mathar_ *)
%t 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
%t (* Third one after Lucas numbers formula *)
%t Table[27/125 (-5 + (-1)^n ((-1)^(n + 1) 6 + LucasL[15 (-1 + 2 n)])), {n, 10}] (* _Artur Jasinski_, Nov 18 2011*)
%Y Cf. A078933, A179107, A179108, A179109, A179387, A179388, A199496.
%K sign
%O 1,1
%A _Artur Jasinski_, Nov 14 2011
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