Minimal graded free resolution for monomial curves in A4 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 gcd(m0,m1,m2,n)=1, such that m0<m1<m2 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[?] of the affine monomial curve parametrically defined by X0=tm0,X1=tm1,X2=tm3,Y=tn is a graded R-module, where R is the polynomial ring k[X0,X1,X3,Y] with the grading degXi:=mi,degY:=n. In this paper, we construct a minimal graded free resolution for k[?].