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 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).

Lights Out, as you mentioned in the linked MSE thread, probably you should be interested in the determinant in $\mathbb{F}_2$, not in $\mathbb{R}$. $\endgroup$ – Federico Poloni Jul 31 '15 at 10:04