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Let $o\left(n\right)$ be the greatest odd integer less than or equal to $n$. In this paper we provide explicit formulae to compute $\mathbb{N}$-graded Betti numbers of the circulant graphs $C_{2n}(1,2,3,5,...,o\left(n\right))$. We do this by showing that this graph is the product (or join) of the cycle $C_n$ by itself, and computing Betti numbers of $C_n*C_n$. We also discuss whether such a graph (more generally, $G*H$) is well-covered, Cohen-Macaulay, sequentially Cohen-Macaulay, Buchsbaum, or $S_2$.