In this paper, we analyze recurrence relations generalized from the Tower of Hanoi problem of the form T (n ,α,β)=min_(1≤t ≤n ){αT (n -t ,α,β)+βS (t ,3)}, where S (t ,3)=2~(t) -1 is the optimal total number of moves for the 3-peg Tower of Hanoi problem. It is shown that when α and β are natural numbers, 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 β2~(i) α~(j) (i ,j ≥0) lined in the increasing order.
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