  Studying Hilbert functions of concrete examples of normal toric rings, it is
demonstrated that, for each $1 \leq s \leq 5$, an $O$-sequence $(h_0, h_1,
\ldots, h_{2s-1}) \in \mathbb{Z}_{\geq 0}^{2s}$ satisfying the properties that
(i) $h_0 \leq h_1 \leq \cdots \leq h_{s-1}$, (ii) $h_{2s-1} = h_0$, $h_{2s-2} =
h_1$ and (iii) $h_{2s - 1 - i} = h_i + (-1)^{i}$, $2 \leq i \leq s - 1$, can be
the $h$-vector of a Cohen--Macaulay standard $G$-domain.
