On generalized Frame-Stewart numbers
Résumé
For the multi-peg Tower of Hanoi problem with $k \ge 4$ pegs, so far the best solution is obtained by the Stewart's algorithm based on the the following recurrence relation: $S_k(n) = \min_{1 \le t \le n} \{ 2 \cdot S_k(n-t) + S_{k-1}(t) \}$, $S_3(n) = 2^n - 1$. In this paper, we generalize this recurrence relation to $G_k(n) = \min_{1 \le t \le n} \{ p_k \cdot G_k(n-t) + q_k \cdot G_{k-1}(t) \}$, $G_3(n) = p_3 \cdot G_3(n-1) + q_3$, for two sequences of arbitrary positive integers $(p_i)_{i \ge 3}$ and $(q_i)_{i \ge 3}$ and we show that the sequence of differences $(G_k(n)-G_k(n-1))_{n \ge 1}$ consists of numbers of the form $(\prod_{i=3}^{k}q_i) \cdot (\prod_{i=3}^{k}{p_i}^{\alpha_i})$, with $\alpha_i \ge 0$ for all $i$, lined in the increasing order. We also apply this result to analyze recurrence relations for the Tower of Hanoi problems on several graphs.
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