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Let $a$ and $d$ be two linearly independent vectors in $\mathbb{N}^2$, overthe field of rational numbers. For a positive integer $k \geq 2$, consider thesequence $a, a+d, \ldots, a+kd$ 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 $k[S_{a,d,k}]$ associatedto this semigroup. We prove that $k[S_{a,d,k}]$ is always Cohen-Macaulay and itis Gorenstein if and only if $k=2$. For $k=2,3,4$, we explicitly compute thesyzygies, 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 definingideal 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 $a,d,k.$

On the algebraic invariants of certain affine semigroup algebras