Another way, using PowerRepresentations[]:
s = Tally[Join @@ (Select[PowersRepresentations[748, #, 2],
Union@# == # && Max@# < 17 && Min@# > 0 &] & /@ Range@8),
Union[#1, #2] == Range@16 &][[All, 1]];
Length@s
(*
114
*)