Squares that are the sum of 4 unequal squares of Fibonacci numbers.
36, 81, 196, 196, 196, 361, 484, 900, 1225, 2401, 3364, 6084, 8100, 15876, 54756, 164025, 373321, 2553604, 17489124, 119836809, 821280964, 5628900676, 38580423561, 264432492900, 1812442912900, 12422657127241, 85146128781156, 583600170521124
1
That is, s^2 = f1^2 + f2^2 + f3^2 + f4^2, where the fi are Fibonacci numbers not all equal. Starting with the 17th number, 611^2 = 373321, the numbers appear to satisfy a linear recurrence, which will be explored in subsequent sequences. All five squares are listed in S000485.
T. D. Noe, Plot of 60 terms
T. D. Noe, Table of 60 terms
Eric W. Weisstein, MathWorld: Fibonacci Number
(Mma) nn = 41; PerfectSquareQ[n_] := JacobiSymbol[n, 13] =!= -1 && JacobiSymbol[n, 19] =!= -1 && JacobiSymbol[n, 17] =!= -1 && JacobiSymbol[n, 23] =!= -1 && IntegerQ[Sqrt[n]]; f2 = Table[Fibonacci[n]^2, {n, 2, 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, f2}, {b, f2}, {c, f2}, {d, f2}]; t
nonn,nice
T. D. Noe, Feb 18 2015