Finite beta-expansions of natural numbers

Let $\beta >1$. For $x\in [0,\mathrm{\infty })$, we have so-called the $\beta$-expansion of $x$ in base $\beta$ as follows: $$x=\sum _{j\le k}{x}_j{\beta }^j=x_k{\beta }^k+\cdots +x_1\beta +x_0+x_{-1}{\beta }^{-1}+x_{-2}{\beta }^{-2}+\cdots$$ where $k\in \mathbb{Z}$, ${\beta }^k\le x<{\beta }^{k+1}$, $x_j\in \mathbb{Z}\cap [0,\beta )$ for all $j\le k$ and ${\sum }_{j\le n}{x}_j{\beta }^j<{\beta }^{n+1}$ for all $n\le k$. In this paper, we give a sufficient condition (for $\beta$) such that each element of $\mathbb{N}$ has the finite beta-expansion in base $\beta$. Moreover we also find a $\beta$ with this finiteness property which does not have positive finiteness property.