  Given an arbitrary field k and an arithmetic sequence of positive integers
m_0<...<m_n, we consider the affine monomial curve parameterized by
X_0=t^{m_0},...,X_n=t^{m_n}. In this paper, we conjecture that the Betti
numbers of its coordinate ring are completely determined by n and the value of
m_0 modulo n. We first show that the defining ideal of the monomial curve can
be written as a sum of two determinantal ideals. Using this fact, we describe
the minimal free resolution of the coordinate ring in the following three
cases: when m_0 is 1 modulo n (determinantal), when m_0 is n modulo n (almost
determinantal), and when m_0 is 2 modulo n and n=4 (Gorenstein of codimension
4).
