(* check if vector x works when rotated k times *)
ok[x_, k_] := Block[{n = Length[x], m, v},
  m = RotateRight[x, #] & /@ Range[k];
  v = Tuples[{0, 1}, n];
  Unequal @@ (m.# & /@ v)];

(* compute a canonical representative for the rotation class of x *)
MinRotation[x_] :=
  First[Sort[RotateRight[x, #] & /@ Range[Length[x]]]];

f[n_] := f[n] = Block[{k, v},
  v = Select[Tuples[{0, 1}, n], # == MinRotation[#] &];
  For[k = 1, k <= n, k++,
  s = Select[v, ok[#, k] &];
  If[s != {}, Return[{k, First[s], Length[s]}]]]];

TableForm[Prepend[f[#], #] & /@ Range[12],
  TableDirections -> {Column, Row, Row},
  TableHeadings -> {None, {"n", "f(n)", "Example", "# of examples"}}]