f[x_] := Abs[Extract[list, x + n] - Extract[list, x]]

g = Length[Cases[Map[f, Table[i, {i, Length[list] - n}]], 0]]

k[list_, dist_] := Count[ListConvolve[Join[{1}, ConstantArray[0, dist - 1], {1}], list, {-1, 1}, 0,
                                      Times, First@# == Last@# &@List@## &],
                        True]

list = {a, e, a, e, b, e, b, c, d};
k[{a, e, a, e, b, e, b, c, d}, #] & /@ Range[Length[list] - 2]

(* {0, 4, 0, 1, 0, 0, 0} *)

Clear[countSameLetters]
countSameLetters[list_List, distance_Integer] :=
 Position[
   Flatten@Differences@Position[list, #] & /@
    CharacterRange["a", "z"], distance] // Length

Clear[countAndDisplaySameLetters]
countAndDisplaySameLetters[list_List, distance_Integer] :=
 Module[{distancePerLetter, aToZ, letterPositions},
  aToZ = CharacterRange["a", "z"];
  letterPositions =
   First /@
    Position[Differences@Flatten@Position[list, #] & /@ aToZ,
     distance];
  {distance, Length[letterPositions],
   Part[aToZ, DeleteDuplicates@letterPositions]}
  ]

randomChar = RandomChoice[CharacterRange["a", "z"], 100]

countSameLetters[randomChar, 5]
(* 2 *)

countAndDisplaySameLetters[randomChar, 5]
(* {5, 2, {"e", "i"}} *)

countAndDisplaySameLetters[randomChar, #] & /@ Range[1, 10] //
 TableForm[#,
   TableHeadings -> {None, {"Distance", "Count", "Letters"}},
   TableSpacing -> {5, 2}] &