On the number of reachable configurations for the chessboard pebbling problem

Recently, Chung, Graham, Morrison and Odlyzko [F. Chung, R. Graham, J. Morrison, A. Odlyzko, Pebbling a chessboard, Amer. Math. Monthly 102 (1995) 113–123] studied various combinatorial and asymptotic enumeration aspects of chessboard pebbling. Here, a pebble is placed at the origin ( 0 , 0 ) of an infinite chessboard. At each step, a pebble is removed from ( i , j ) and replaced by two pebbles at positions ( i , j + 1 ) and ( i + 1 , j ) (provided these are unoccupied). After k steps, the board has k + 1 pebbles in various arrangements. Here we study the number of possible arrangements asymptotically, as k → ∞ . We analyze the recurrence derived in [F. Chung, R. Graham, J. Morrison, A. Odlyzko, Pebbling a chessboard, Amer. Math. Monthly 102 (1995) 113–123] by methods of applied mathematics, such as WKB expansions and matched asymptotics. In particular, we obtain an analytic expression for the growth rate of the number of possible arrangements.