If we have a triangular matrix we can calculate the determinant in $O(n)$. If we have a triangular matrix with one extra diagonal above the main diagonal, so for example:
\begin{Vmatrix} a_1 & a_2 & 0 & 0 & 0 & 0 \\ b_1 & b_2 & b_3 & 0 & 0 & 0 \\ c_1 & c_2 & c_3 & c_4 & 0 & 0 \\ d_1 & d_2 & d_3 & d_4 & d_5 & 0 \\ e_1 & e_2 & e_3 & e_4 & e_5 & e_6 \\ f_1 & f_2 & f_3 & f_4 & f_5 & f_6 \end{Vmatrix}
where we have the extra diagonal $[a_2, b_3, c_4,...,f_6]$, is there a way to calculate the determinant faster than the standard $O(n^3)$?