Exact Analysis of the Recurrence Relations Generalized from the Tower of Hanoi

In this paper, we analyze the recurrence relations generalized from the Tower of Hanoi problem of the form T (n, α, β) = min1≤t≤n{αT (n− t, α, β)+β S(t, 3)}, where S(t, 3) = 2−1 is the optimal solution for the 3-peg Tower of Hanoi problem. It is shown that when α and β are natural numbers and α ≥ 2, the sequence of differences of T (n, α, β)’s, i.e., T (n, α, β) − T (n − 1, α, β), consists of numbers of the form β2α (i, j ≥ 0) lined in the increasing order.