Let $[0]_q:=1$ and $[n]_q:=\frac{1-q^n}{1-q}=1+q+\cdots+q^{n-1}$, for $n\geq1$.

Question. Is there a closed formula (with proof) for the determinant of the matrix of $(i,j)$-entries $$[i+j\bmod n]_q, \qquad i,j=1,2,\dots,n.$$

Remark. To bring in some context to the problem, this determinant is the specialization $$x_{i+j\bmod n}\rightarrow [i+j\bmod n]_q$$ in the group determinant of Frobenius for the (finite) additive group $\mathbb{Z}_n$.