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 $A$? 

Wang et. al [[1]] show 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.

**References**

  [[1]] W. Wang, C. M. Wang and S. L. Guo, [On the walk matrix of the Dynkin graph $D_n$][1], Linear Algebra Appl. 653 (2022) 193-206.


  [1]: https://www.sciencedirect.com/science/article/pii/S0024379522002920?via%3Dihub