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{α T(n − t, α, β)+β S(t, 3)}, where S(t, 3) = 2t − 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 β2iαj (i, j ≥ 0) lined in the increasing order.