It is while studying the Hückel Method of Physical Chemistry that I came across the following recurrence relation: \begin{align*} U_n(x)=xU_{n-1}(x)-U_{n-2}(x)+(-1)^{n-1}(4+2x) \end{align*} Where \begin{align*} U_n(x):=\underbrace{\left|\begin{matrix} x& 1 &&&&1\\ 1 & x& 1 &&& \\ & 1 & x &1&& \\ && 1 &x&& \\ &&&& \ddots &1\\ 1 &&&&1&x \end{matrix}\right|}_n \end{align*}
For the related determinant $\displaystyle D_n(x):=\underbrace{\left|\begin{matrix} x& 1 &&&&\\ 1 & x& 1 &&& \\ & 1 & x &1&& \\ && 1 &x&& \\ &&&& \ddots &1\\ &&&&1&x \end{matrix}\right|}_n$, we have the nicer recurrence relation: \begin{align*} D_n(x)=xD_{n-1}(x)-D_{n-2}(x) \end{align*} which is a Chebyshev polynomial of the first kind.
However, I would like to somehow find a closed form for $U_n(x)$. Is this at all possible?