On determinants involving second-order recurrent sequences

Let $A$ and $B$ be complex numbers, and let $(w_n{)}_{n\ge 0}$ be a sequence of complex numbers with $w_{n+1}=A{w}_n-B{w}_{n-1}$ for all $n=1,2,3,\dots$. When $w_0=0$ and $w_1=1$, the sequence $(w_n{)}_{n\ge 0}$ is just the Lucas sequence $(u_n(A,B){)}_{n\ge 0}$. In this paper, we evaluate the determinants $$\det [w_{\mathrm{\mid }j-k\mathrm{\mid }}{]}_{1\le j,k\le n}\text{ and }\det [w_{\mathrm{\mid }j-k+1\mathrm{\mid }}{]}_{1\le j,k\le n}\mathrm{.}$$ In particular, we have $$\det [u_{\mathrm{\mid }j-k\mathrm{\mid }}(A,B){]}_{1\le j,k\le n}=(-1{)}^{n-1}{u}_{n-1}(2A,(B+1{)}^2)\mathrm{.}$$ When $B=-1$ and $2\mid n$, we also determine the characteristic polynomial of the matrix $[w_{j+k}{]}_{0\le j,k\le n-1}$.