Wythoff's sequence and N-Heap Wythoff's conjectures

Define a Wythoff's sequence as a sequence of pairs of integers {(A(n), B-n)}(n > n0) such that there exists a finite set of integers T, A(n) = mex ({A(i), B-i : i < n} boolean OR T), B-n - A(n) = n, and {B-n} boolean AND T = theta. Structural properties and behaviors of Wythoff's sequence are investigated. The main result is that for such a sequence, there always exists an integer alpha such that when n is large enough, vertical bar A(n) - vertical bar n phi vertical bar - alpha vertical bar <= 1, where phi = (1 + root 5)/2, the golden section. The value of alpha can also be easily determined by a relatively small number of pairs in the sequence. As a corollary, the two conjectures on the N-heap Wythoff's game by Fraenkel [Complexity, appeal and challenges of combinatorial Games, Theoret. Comput. Sci. 313 (2004) 393-415] on the N-heaped Wythoff's game are proved to be equivalent. (c) 2005 Elsevier B.V. All rights reserved.