## S000974

Irregular table of solutions d to the Pell equation x^2 - d*y^2 = -1, where x is tribonacci number t(n).

5, 17, 2, 50, 170, 577, 1937, 6562, 22202, 75077, 254017, 859330, 2907026, 9834497, 1330793, 33269825, 112550882, 380757170, 1288092101, 4357584145, 14741602226, 49870482490, 168710633537, 570743986577, 1930813074370, 261275753722, 6531893843050

1

T. D. Noe, Plot of 136 terms

T. D. Noe, Table of 136 terms

Florian Luca, Amanda Montejano, Laszlo Szalay, and Alain Togbe, On the x-coordinates of Pell equations which are tribonacci numbers, arXiv 1612.09546 (Dec 30 2016)

(Mma) nn = 64; s1 = Table[0, {nn}]; s2 = s1; tri = {0, 0, 1, 1}; t = Table[AppendTo[tri, tri[[-1]] + tri[[-2]] + tri[[-3]]]; x = tri[[-1]]; s1[[n]] = FindInstance[x^2 - d*y^2 == 1 && d > 1 && y > 0, {d, y}, Integers, 20]; s2[[n]] = FindInstance[x^2 - d*y^2 == -1 && d > 1 && y > 0, {d, y}, Integers, 20]; s1[[n]] = Table[{s1[[n, i, 1, 2]], s1[[n, i, 2, 2]]}, {i, Length[s1[[n]]]}]; s2[[n]] = Table[{s2[[n, i, 1, 2]], s2[[n, i, 2, 2]]}, {i, Length[s2[[n]]]}]; {Length[s1[[n]]], Length[s2[[n]]]}, {n, 4, nn}]; Flatten[Table[Transpose[s2[[n]]][[1]], {n, 4, nn}]]

Cf. S000970.

nonn

T. D. Noe, Feb 24 2017

© Tony D Noe 2014-2017