Minimal graded free resolutions for monomial curves in A4 defined by almost arithmetic sequences

Let m = (m<inf>0</inf>, m<inf>1</inf>, m<inf>2</inf>, n) be an almost arithmetic sequence, i.e., a sequence of positive integers with gcd(m<inf>0</inf>, m<inf>1</inf>, m<inf>2</inf>, n) = 1, such that m<inf>0</inf> < m<inf>1</inf> < m<inf>2</inf> form an arithmetic progression, n is arbitrary and they minimally generate the numerical semigroup Γ =m<inf>0</inf>ℕ +m<inf>1</inf>ℕ +m<inf>2</inf>ℕ +nℕ. Let k be a field. The homogeneous coordinate ring k[Γ] of the affine monomial curve parametrically defined by X<inf>0</inf> = t^{m<inf>0</inf>}, X<inf>1</inf> = t^{m<inf>1</inf>}, X<inf>2</inf> = t^{m<inf>2</inf>}, Y = tⁿ is a graded R-module, where R is the polynomial ring k[X<inf>0</inf>, X<inf>1</inf>, X<inf>2</inf>, Y] with the grading degX<inf>i</inf>: = m<inf>i</inf>, degY: = n. In this paper, we construct a minimal graded free resolution for k[Γ].