According to doi:10.1016/S0024-3795(99)00114-7, the closed form for the eigenvalues of a tridiagonal Toepliz matrix of the form

$$ \begin{bmatrix}a & b\\ c & a & b\\ & \ddots & a & \ddots \\ & & & \ddots & \\ & & & c & a \end{bmatrix} $$

is:

$$\lambda_{k}=a+2\sqrt{bc}\cos\left[\frac{k\pi}{(n+1)}\right], \quad k=1\cdots n $$

According to doi:10.1016/S0024-3795(99)00114-7, the closed form for the eigenvalues of a tridiagonal Toepliz matrix of the form

$$ \begin{bmatrix}a & b\\ c & a & b\\ & \ddots & a & \ddots \\ & & & \ddots & \\ & & & c & a \end{bmatrix} $$

is:

$$\lambda_{k}=a+2\sqrt{bc}\cos\left[\frac{k\pi}{(n+1)}\right], \quad k=1\cdots n $$