  A generalized Beatty sequence is a sequence $V$ defined by
$V(n)=p\lfloor{n\alpha}\rfloor+qn +r$, for $n=1,2,\dots$, where $\alpha$ is a
real number, and $p,q,r$ are integers. These occur in several problems, as for
instance in homomorphic embeddings of Sturmian languages in the integers. Our
results are for the case that $\alpha$ is the golden mean, but we show how some
results generalise to arbitrary quadratic irrationals. We mainly consider the
following question: For which sixtuples of integers $p,q,r,s,t,u$ are the two
sequences $V=(p\lfloor{n\alpha}\rfloor+qn +r)$ and
$W=(s\lfloor{n\alpha}\rfloor+tn +u)$ complementary sequences?
  We also study complementary triples, i.e., three sequences
$V_i=(p_i\lfloor{n\alpha}\rfloor+q_in+r_i), \:i=1,2,3$, with the property that
the sets they determine are disjoint with union the positive integers.
