I think this is false.

Take the first row $A=(1,-1,1,-1,1,0,0)$.

The circulant matrix is:
$$
\left(\begin{array}{rrrrrrr}
1 & -1 & 1 & -1 & 1 & 0 & 0 \\
0 & 1 & -1 & 1 & -1 & 1 & 0 \\
0 & 0 & 1 & -1 & 1 & -1 & 1 \\
1 & 0 & 0 & 1 & -1 & 1 & -1 \\
-1 & 1 & 0 & 0 & 1 & -1 & 1 \\
1 & -1 & 1 & 0 & 0 & 1 & -1 \\
-1 & 1 & -1 & 1 & 0 & 0 & 1
\end{array}\right)
$$

The determinant is $1$.

Your definition is with columns, so you may need
to transpose.

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Experimentally first row starting $(1,-1,1)$ followed by $n$ zeros
with determinant $\pm 1$
is [A047235](https://oeis.org/A047235)  `Numbers that are congruent to {2, 4} mod 6`