Abstract
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).
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