Patterns in the generalized Fibonacci word, applied to games

We analyze a natural generalization, W, of the infinite Fibonacci word over the alphabet Σ={a,b}. We provide tools to represent explicitly the set {s∈Z≥0:W(s)=b,W(s+x)=a} for any fixed positive integer x. We show how this representation can be used to analyze the preservation of P-positions of any game whose P-positions are a pair of complementary Beatty sequences, in particular a certain generalization of Wythoff Nim (Holladay, 1968; Fraenkel, 1982).