Intersection of $\{ [n\sqrt{2}]\mid n \in \mathbb{N}^* \}$ and $\{ [n(2+\sqrt{2})]\mid n \in \mathbb{N}^* \}$ Find the intersection of sets $A$ and $B$ where
$$A = \{ [n\sqrt{2}]\mid n \in \mathbb{N}^* \}$$
$$B = \{ [n(2+\sqrt{2})]\mid n \in \mathbb{N}^* \}.$$

([$x$] is the integer part of $x$)

Using the computer, we found common elements.

Does anyone have an idea to solve?
These are known as Beatty sequences and they've been quite thoroughly studied (you can find one of your two sequences at http://oeis.org/A001951 ).  The most interesting property (and most relevant here) is that if $r$ and $s$ are irrational numbers with $r,s \gt 1$ and $\frac1r+\frac1s=1$, then the sequences $R=\left\{\lfloor (nr)\rfloor\right\}$ and $S=\left\{\lfloor (ns)\rfloor\right\}$ are complementary: $R\cap S=\emptyset$ and $R\cup S=\mathbb{N}$.
Their intersection is the empty set