Number of solutions to the Pell equation x^2 - d*y^2 = -1 or 1, where x is tribonacci number t(n).
2, 2, 5, 3, 3, 3, 4, 5, 3, 2, 5, 4, 2, 4, 4, 7, 2, 2, 4, 3, 2, 3, 5, 4, 2, 2, 6, 10, 2, 3, 9, 4, 3, 3, 7, 3, 2, 3, 7, 3, 3, 3, 5, 4, 5, 2, 4, 4, 2, 2, 4, 3, 5, 2, 9, 3, 2, 3, 10, 5, 2, 3, 5, 5, 2, 3, 7, 5, 5, 3, 4, 5, 3, 3, 5, 4, 3, 3, 4, 7, 2, 2, 13, 9, 3, 2, 5
4
The tribonacci numbers satisfy t(0) = t(1) = 0, t(2) = 1, and t(k+3) = t(k) + t(k+1) + t(k+2). So the tribonacci numbers begin 0, 0, 1, 1, 2, 4, 7, 13, 24, 44. The 1 and -1 cases are treated separately in sequences S000971 to S000974.
T. D. Noe, Plot of terms 4..128
T. D. Noe, Table of terms 4..128
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}]; Table[t[[n, 1]] + t[[n, 2]], {n, nn - 3}]
nonn
T. D. Noe, Feb 24 2017