  Let $\alpha$ and $\beta$ belong to the same quadratic field. We show that the
inhomogeneous Beatty sequence $(\lfloor n \alpha + \beta \rfloor)_{n \geq 1}$
is synchronized, in the sense that there is a finite automaton that takes as
input the Ostrowski representations of $n$ and $y$ in parallel, and accepts if
and only if $y = \lfloor n \alpha + \beta \rfloor$. Since it is already known
that the addition relation is computable for Ostrowski representations based on
a quadratic number, a consequence is a new and rather simple proof that the
first-order logical theory of these sequences with addition is decidable. The
decision procedure is easily implemented in the free software Walnut.
  As an application, we show that for each $r \geq 1$ it is decidable whether
the set $\{ \lfloor n \alpha + \beta \rfloor \, : \, n \geq 1 \}$ forms an
additive basis (or asymptotic additive basis) of order $r$. Using our
techniques, we also solve some open problems of Reble and Kimberling, and give
an explicit characterization of a sequence of Hildebrand et al.
