ClearAll[Fusc]
Fusc[0] := 0
Fusc[1] := 1
Fusc[n_] := Fusc[n] = If[EvenQ[n], Fusc[n/2], Fusc[(n - 1)/2] + Fusc[(n + 1)/2]]
Fusc /@ Range[0, 60]
res = {{0, 1}};
i = 0;
PrintTemporary[Dynamic[{res, i}]];
While[Length[res] < 6,
  f = Fusc[i];
  If[IntegerLength[res[[-1, -1]]] < IntegerLength[f],
   AppendTo[res, {i, f}]
   ];
  i++;
  ];
res