The question is the following: given a matrix
$$A=\begin{pmatrix}
  1& 2 &  &  &  & \\
  1&  0& 1 &  & & \\
  &  1&  0& 1 &  &\\
   &  & \ddots & \ddots  & \ddots & \\
   &  &  & 1& 0 & 1\\
   &  &  &  & 1 &0
\end{pmatrix}.$$
Is it possible to give analytic expressions for the eigenvalues and eigenvectors of $A$? 

It has been shown that if the elements on the main diagonal are all 0, the eigenvalues and eigenvectors of $A$ can be expressed in trigonometric functions.

Thanks for your answer.