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 in [15], based on the following recurrence relation: S k ( n ) = min 1 ≤ t ≤ n { 2 ⋅ 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 ≤ 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 = 3 k q i ) ⋅ ( ∏ i = 3 k p i α 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.