On the algebraic invariants of certain affine semigroup algebras

Let $a$ and $d$ be two linearly independent vectors in $\mathbb{N}^2$, over the field of rational numbers. For a positive integer $k \geq 2$, consider the sequence $a, a+d, \ldots, a+kd$ such that the affine semigroup $S_{a,d,k} = \langle a, a+d, \ldots, a+kd \rangle$ is minimally generated by this sequence. We study the properties of affine semigroup algebra $k[S_{a,d,k}]$ associated to this semigroup. We prove that $k[S_{a,d,k}]$ is always Cohen-Macaulay and it is Gorenstein if and only if $k=2$. For $k=2,3,4$, we explicitly compute the syzygies, minimal graded free resolution and Hilbert series of $k[S_{a,d,k}].$ We also give a minimal generating set and a Gr\"{o}bner basis of the defining ideal of $k[S_{a,d,k}].$ Consequently, we prove that $k[S_{a,d,k}]$ is Koszul. Finally, we prove that the Castelnuovo-Mumford regularity of $k[S_{a,d,k}]$ is $1$ for any $a,d,k.$