A Generalization of Wythoff's Game

Wythoff s game is a variation of Nim, a two-pile game in which each player removes counters in turn until the winner takes the last counter. The safe-pairs generated in the solution of Wythoff's game have many properties interesting in themselves, and are related to the canonical Zeckendorf representation of an integer using Fibonacci numbers. In Nim, the strategy is related to expressing the numbers in each pile in binary notation, or representing them by powers of 2. Here, the generalized game provides number sequences related to the canonical Zeckendorf representation of integers using Lucas numbers.