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.
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