matrices - Determinant of a symmetric Toeplitz matrix

Let $A_n = (a_{ij})$ be an $n\times n$ matrix such that $a_{ii} = 0, a_{ij} = 2$ when $|j − i| = 1$, and $a_{ij} = 1$ otherwise. The question is to find the determinant in terms of $n$. I computed the first six terms, depending on $n$, but, unfortunately, no clear relationship was found. Here they are:
\begin{align}

\det A_1&=0,\\
\det A_2&=-4,\\
\det A_3&=8,\\
\det A_4&=-7,\\
\det A_5&=0,\\
\det A_6&=7.
\end{align}
Laplace expansion turned out to be useful only if $n$ is known. How can one derive a formula for $\det A_n$ in terms of $n$?