Is there a formula to compute the determinant of block tridiagonal matrices when the determinants of the involved matrices are known? 

In particular, I am interested in the case

$$A = \begin{pmatrix} J_n & I_n & 0 & \cdots & \cdots & 0 \\ I_n & J_n & I_n & 0 & \cdots & 0 \\ 0 & I_n & J_n  & I_n & \ddots & \vdots \\ \vdots & \ddots & \ddots & \ddots &  \ddots & 0 \\ 0 & \cdots & \cdots & I_n & J_n & I_n \\ 0 & \cdots & \cdots & \cdots & I_n & J_n \end{pmatrix}$$

where $J_n$ is the $n \times n$ tridiagonal matrix whose entries on the sub-, super- and main diagonals are all equal to $1$ and $I_n$ is identity matrix of size $n$.

I have [asked this question before][1] on MathStackExchange, where a user came up with an algorithm. Nevertheless, I am interested if there is an explicit formula (or at least, if one can say in which cases the determinant is nonzero).


  [1]: https://math.stackexchange.com/q/1307671