  Let $a, b$ and $n > 1$ be three positive integers such that $a$ and
$\sum_{j=0}^{n-1} b^j$ are relatively prime. In this paper, we prove that the
toric ideal $I$ associated to the submonoid of $\mathbb{N}$ generated by
$\{\sum_{j=0}^{n-1} b^j\} \cup \{\sum_{j=0}^{n-1} b^j + a\, \sum_{j=0}^{i-2}
b^j \mid i = 2, \ldots, n\}$ is determinantal. Moreover, we prove that for $n >
3$, the ideal $I$ has a unique minimal system of generators if and only if $a <
b-1$.
