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 \(k[\Gamma]\).