  Let $k$ be an arbitrary field, the purpose of this work is to provide
families of positive integers $\mathcal{A} = \{d_1,\ldots,d_n\}$ such that
either the toric ideal $I_{\mathcal A}$ of the affine monomial curve $\mathcal
C = \{(t^{d_1},\ldots,\,t^{d_n}) \ | \ t \in k\} \subset \mathbb{A}_k^n$ or the
toric ideal $I_{\mathcal A^{\star}}$ of its projective closure ${\mathcal
C^{\star}} \subset \mathbb{P}_k^n$ is a complete intersection. More precisely,
we characterize the complete intersection property for $I_{\mathcal A}$ and for
$I_{\mathcal A^{\star}}$ when:
  (a) $\mathcal{A}$ is a generalized arithmetic sequence,
  (b) $\mathcal{A} \setminus \{d_n\}$ is a generalized arithmetic sequence and
$d_n \in \mathbb{Z}^+$,
  (c) $\mathcal{A}$ consists of certain terms of the $(p,q)$-Fibonacci
sequence, and
  (d) $\mathcal{A}$ consists of certain terms of the $(p,q)$-Lucas sequence.
  The results in this paper arise as consequences of those in Bermejo et al.
[J. Symb. Comput. 42 (2007)], Bermejo and Garc\'{\i}a-Marco [J. Symb. Comput.
(2014), to appear] and some new results regarding the toric ideal of the curve.
