  Let $\mm=(m_0,m_1,m_2,n)$ be an almost arithmetic sequence, i.e., a sequence
of positive integers with ${\rm gcd}(m_0,m_1,m_2,n) = 1$, such that
$m_0<m_1<m_2$ form an arithmetic progression, $n$ is arbitrary and they
minimally generate the numerical semigroup $\Gamma = m_0\N + m_1\N + m_2\N +
n\N$. Let $k$ be a field. The homogeneous coordinate ring $k[\Gamma]$ of the
affine monomial curve parametrically defined by
$X_0=t^{m_0},X_{1}=t^{m_1},X_2=t^{m_3},Y=t^{n}$ is a graded $R$-module, where
$R$ is the polynomial ring $k[X_0,X_1,X_3, Y]$ with the grading
$\deg{X_i}:=m_i, \deg{Y}:=n$. In this paper, we construct a minimal graded free
resolution for $k[\Gamma]$.
