Enumerating 4x4 "Android-like" pattern locks

validPatternQ[pattern_List, n_] :=
 Module[
  {k},
  ! (Or @@ Table[
      MemberQ[
       Drop[pattern, k],
       Alternatives @@ (
         {{1/2, 1/2}, {1/3, 2/3}, {2/3, 1/3}}.pattern[[{k - 1, k}]])
       ],
      {k, 2, n}
      ])
  ]
size = 4;
points = Tuples[Range[size], 2];
Sum[
 {n, k} = {1000000, 0};
 Do[If[validPatternQ[RandomSample[points], len], ++k], {n}];
 (size^2)!/(size^2 - len)!*k/n,
 {len, 4, size^2}
]