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 $n$?