On generalized Frame-Stewart numbers

For the multi-peg Tower of Hanoi problem with k ≥ 4 pegs, so far the best solution is obtained by the Stewart’s algorithm [15] based on the the following recurrence relation: Sk(n) = min 1≤t≤n � 2 · Sk(n− t) + Sk−1(t) � , S3(n) = 2 − 1. In this paper, we generalize this recurrence relation to Gk(n) = min 1≤t≤n � pk · Gk(n− t) + qk · Gk−1(t) � , G3(n) = p3 · G3(n− 1) + q3, for two sequences of arbitrary positive integers (pi)i≥3 and (qi)i≥3 and we show that the sequence of differences (Gk(n) − Gk(n − 1))n≥1 consists of numbers of the form ( �k i=3 qi) · ( �k i=3 pi αi), with αi ≥ 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.