Let $W_{alpha}$ be a weighted shift with positive weight sequence ${alpha}={alpha_i}_{i=0}^{infty}$. The semi-cubical hyponormality of $W_{alpha}$ is introduced and some flatness properties of $W_{alpha}$ are discussed in this note. In particular, it is proved that if ${alpha}_n={alpha}_{n+1}$ for some $n{geq}1$, ${{alpha}_{n+k}}={alpha}_n$ for all $k{geq}1$.