Enumerating moves in the optimal solution of the Tower of Hanoi

In the Tower of Hanoi problem, there is six types of moves between the three pegs. The main purpose of the present paper is to find out the number of each of these six elementary moves in the optimal sequence of moves. We present a recursive function based on indicator functions, which counts the number of each elementary move, we investigate some of its properties including combinatorial identities, recursive formulas and generating functions. Also we found and interesting sequence that is strongly related to counting each type of these elementary moves that we'll establish some if its properties as well.