Determinant of huge block matrix

Question

I need to calculate the determinant of the $3j \times 3j$ symmetric block matrix $$\mathsf A_{j} = \begin{bmatrix} \mathsf T_j & \mathsf V_j & \mathsf 0_j \\ -\mathsf V_j & \mathsf T_j & \mathsf 0_j \\ \mathsf 0_j & \mathsf 0_j & \mathsf T_j \end{bmatrix},$$ where $$\mathsf T_j = \begin{bmatrix} 2 & -1 & & \\ -1 & 2 & -1 & \\ & -1 & 2 & \\ & & & \ddots \end{bmatrix}, \qquad \mathsf V_j = \begin{bmatrix} 0 & \eta & & \\ -\eta & 0 & \eta & \\ & -\eta & 0 & \\ & & & \ddots \end{bmatrix}, \quad \eta \in \mathbb R. $$ I have managed to calculate $\det \mathsf T_j = j+1$, so that $$ \det \mathsf A_j = (j+1) \det \begin{bmatrix} \mathsf T_j & \mathsf V_j \\ -\mathsf V_j & \mathsf T_j \end{bmatrix},$$ but the last determinant seems way too difficult to evaluate. I would be satisfied with a recurrence relation for the sequence $a_j := \det \mathsf A_j$, but that seems to be just as difficult.