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 β2ⁱα^j (i,j≥0) lined in the increasing order.