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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 thetoric 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 $a <b-1$.

Minimal binomial systems of generators for the ideals of certain monomial curves