Generalized Beatty sequences and complementary triples
Jean‐Paul Allouche, F. Michel Dekking · Moscow Journal of Combinatorics and Number Theory · 2019
A generalized Beatty sequence is a sequence V defined by V (n) = p nα + qn + r , for n = 1, 2, . . ., where α is a real number, and p, q, r 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 ( p, q, r ), (s, t, u) such that the two sequences V (n) = ( p nα + qn + r ) and W (n) = (s nα + tn + u) are complementary (their image sets are disjoint and cover the positive integers).Most of our results are for the case that α is the golden mean, but we show how some of them generalize to arbitrary quadratic irrationals.We also study triples of sequences V i = ( p i nα + q i n + r i ), i = 1, 2, 3 that are complementary in the same sense.