I am interested in the calculation of the determinant of the $N\times N$ symmetric matrix \begin{equation*} \mathbf B = \left(\begin{array}{*{20}c} 2 & & -1& &-1& &\\ & 2 & & -1& & \ddots& \\ -1& & \ddots& &\ddots & & -1 \\ & -1& & & &-1& \\ -1& & \ddots& &2 &&-1 \\ &\ddots & &-1 & &2& \\ & &-1 & &-1 & &2 \\ \end{array}\right) \end{equation*} where the first $-1$ on the left in the first line from left to right is at position n, and the second $-1$ at position $m>n$. The blank parts of the matrix are implicitly filled with zeros.
Are there results or properties for such class of matrices that would allow to compute the determinant for any dimension $N$, without its explicit calculation (that could be accomplished e.g. via Leibniz formula for determinants)?