Square root of the numbers in S000491.
5, 7, 10, 13, 30, 50, 77, 174
1
This sequence is finite.
T. D. Noe, Plot of 8 terms
Eric W. Weisstein, MathWorld: Lucas Number
(Mma) nn = 51; PerfectSquareQ[n_] := JacobiSymbol[n, 13] =!= -1 && JacobiSymbol[n, 19] =!= -1 && JacobiSymbol[n, 17] =!= -1 && JacobiSymbol[n, 23] =!= -1 && IntegerQ[Sqrt[n]]; l2 = Table[LucasL[n]^2, {n, 0, nn}]; t = {}; Do[If[a >= b >= c >= d && (a != b || a != c || a != d || b != c || b != d || c != d), n = a + b + c + d; If[PerfectSquareQ[n], AppendTo[t, n]]], {a, l2}, {b, l2}, {c, l2}, {d, l2}]; Sqrt[t]
nonn,fini,full
T. D. Noe, Feb 24 2015