Map[
  (
    {n, m} = #;
    m (Map[
        (Times @@ Binomial[4, #]) Binomial[13 - n,
           13 - Total[#]] 4^(13 - Total[#]) &,
        Tuples[{2, 3, 4}, n]
        ] // Total)
    ) &,
  Length /@ Select[
     Union /@ (Select[
          Subsets[Range[14]],
          MatchQ[Differences[#], {___, 1, 1, ___}] &
          ] /. {14 -> 1}) // Union,
     Length[#] < 7 &
     ] // Tally
  ] // Total