Minimal Graded Free Resolution for Monomial Curves in mathbb{A}⁴ defined by almost arithmetic sequences

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 $k[\Gamma]$.