  It is unknown if an Artinian level O-sequence of codimension 3 and type $r
(\ge 2)$ is unimodal, while it is known that any Gorenstein O-sequence of
codimension 3 is unimodal. We show that some Artinian non-unimodal O-sequence
of codimension 3 cannot be level. We also find another non-level case: if some
Artinian algebra $A$ of codimension 3 has the Hilbert function
$$\begin{matrix}\H & : & h_0 & h_1 & ... & h_{d-1} & \underbrace{h_d ...
h_d}_{s\text{-times}} & h_{d+s}, \end{matrix} $$ such that $h_d<h_{d+s}$ and
$s\ge 2$, then $A$ has a socle element in degree $d+s-2$, that is, $A$ is not
level.
