For the multi-peg Tower of Hanoi problem with k⩾4 pegs, so far
the best solution is obtained by the Stewart's algorithm based on the the
following recurrence relation: S_k(n)=min_1⩽t⩽n{2⋅S_k(n−t)+S_k−1(t)},
S_3(n)=2n−−1. In this paper, we generalize this recurrence
relation to G_k(n)=min_1⩽t⩽n{p_k⋅G_k(n−t)+q_k⋅G_k−1(t)},
G_3(n)=p_3⋅G_3(n−1)+q_3, for two sequences of
arbitrary positive integers (p_i)_i⩾3 and
(q_i)_i⩾3 and we show that the sequence of
differences (G_k(n)−G_k(n−1))_n⩾1 consists of numbers of the form (∏_i=3kq_i)⋅(∏_i=3kp_iα_i), with α_i⩾0
for all i, arranged in nondecreasing order. We also apply this result to
analyze recurrence relations for the Tower of Hanoi problems on several graphs.Comment: 13 pages ; 3 figure