A Note on Iterated Beatty Sequences

For any irrational number $\alpha >\frac{3+\sqrt{5}}{2}\approx 2.618$ and given a positive $n\in \mathbb{N}$, we use elementary number theory to introduce a necessary and sufficient condition for a natural number $x$ to be in the $n$th iterate of the Beatty sequence of modulus $\alpha$.