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 $a < b-1$.