A generalized Beatty sequence is a sequence
$V$ defined
by
$V\left(n\right)=p\lfloor n\alpha \rfloor +qn+r$, 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
$V\left(n\right)=\left(p\lfloor n\alpha \rfloor +qn+r\right)$
and
$W\left(n\right)=\left(s\lfloor n\alpha \rfloor +tn+u\right)$
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
${V}_{i}=\left({p}_{i}\lfloor n\alpha \rfloor +{q}_{i}n+{r}_{i}\right)$,
$i=1,2,3$ that
are complementary in the same sense.
