A generalized Beatty sequence is a sequence defined by , for , where is a real number, and are integers. Such sequences occur, for instance, in homomorphic embeddings of Sturmian languages in the integers.
We consider the question of characterizing pairs of integer triples such that the two sequences and are complementary (their image sets are disjoint and cover the positive integers). Most of our results are for the case that is the golden mean, but we show how some of them generalize to arbitrary quadratic irrationals.
We also study triples of sequences , that are complementary in the same sense.
"Generalized Beatty sequences and complementary triples." Mosc. J. Comb. Number Theory 8 (4) 325 - 341, 2019. https://doi.org/10.2140/moscow.2019.8.325