Irregular table of solutions d to the Pell equation x^2 - d*y^2 = 1, where x is tribonacci number t(n).
3, 15, 3, 12, 48, 42, 168, 23, 575, 215, 1935, 410, 1640, 6560, 222, 888, 5550, 22200, 3003, 75075, 254015, 13427, 53708, 214832, 859328, 181689, 726756, 2907024, 9834495, 3696647, 33269823, 7034430, 28137720, 112550880, 2644147, 10576588, 23797323, 42306352
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T. D. Noe, Plot of 380 terms
T. D. Noe, Table of 380 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[s1[[n]]][[1]], {n, 4, nn}]]
Cf. S000970.
nonn
T. D. Noe, Feb 24 2017