Square root of the numbers in S000495.

4, 5, 6, 7, 8, 12, 14, 16, 19, 20, 24, 27, 32, 36, 39, 52, 79, 81, 82, 93, 94, 109, 111, 124, 140, 174, 204, 234, 236, 246, 328, 353, 379, 644, 844, 852, 994, 1407, 1686, 3708, 4414, 5779, 11556, 30254, 39604, 79206, 207364, 271444, 542886, 1421294, 1860499

1

This sequence, as opposed to the 4-square sequence in S000491, is infinite. The infinite part, which becomes obvious starting at the 41-st term, consists of three linear recursive sequences that are explored in subsequent sequences.

T. D. Noe, Plot of 67 terms

T. D. Noe, Table of 67 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}]; t5 = {}; Do[If[a >= b >= c >= d >= e && (a != b || a != c || a != d || a != e || b != c || b != d || b != e || c != d || c != e || d != e), n = a + b + c + d + e; If[PerfectSquareQ[n], AppendTo[t5, n]]], {a, l2}, {b, l2}, {c, l2}, {d, l2}, {e, l2}]; Sqrt[t5]

nonn

T. D. Noe, Feb 24 2015