#### Vol. 8, No. 4, 2019

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Generalized Beatty sequences and complementary triples

### Jean-Paul Allouche and F. Michel Dekking

Vol. 8 (2019), No. 4, 325–341
##### Abstract

A generalized Beatty sequence is a sequence $V$ defined by $V\left(n\right)=p⌊n\alpha ⌋+qn+r$, for $n=1,2,\dots \phantom{\rule{0.3em}{0ex}}$, where $\alpha$ 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 $\left(p,q,r\right),\phantom{\rule{0.3em}{0ex}}\left(s,t,u\right)$ such that the two sequences $V\left(n\right)=\left(p⌊n\alpha ⌋+qn+r\right)$ and $W\left(n\right)=\left(s⌊n\alpha ⌋+tn+u\right)$ are complementary (their image sets are disjoint and cover the positive integers). Most of our results are for the case that $\alpha$ is the golden mean, but we show how some of them generalize to arbitrary quadratic irrationals.

We also study triples of sequences ${V}_{i}=\left({p}_{i}⌊n\alpha ⌋+{q}_{i}n+{r}_{i}\right)$, $i=1,2,3$ that are complementary in the same sense.

##### Keywords
generalized Beatty sequences, complementary pairs and triples, morphic words, return words, Kimberling transform
##### Mathematical Subject Classification 2010
Primary: 11B83, 11B85, 11D09, 11J70, 68R15
##### Milestones
Received: 28 December 2018
Revised: 22 May 2019
Accepted: 5 June 2019
Published: 11 October 2019
##### Authors
 Jean-Paul Allouche CNRS, IMJ-PRG, Sorbonne Université Paris France F. Michel Dekking Delft University of Technology Faculty EEMCS Delft Netherlands