Tridiagonal matrix w/trigonometric eigenvalues

Question

Let $n$ be a natural number and $B$ be the $n\times n$ square matrix of $0$'s and $1$'s $$ B=\begin{pmatrix} 0 & 1 & 0 & \ldots & 0 \\ 1 & 0 & 1 & \ldots & 0 \\ 0 & 1 & \ddots & \ddots & \vdots \\ \vdots & \vdots & \ddots & 0 & 1 \\ 0 & 0 & \ldots & 1 & 1 \\ \end{pmatrix}. $$ Is there a way to find the $n\times n$ diagonal matrix D, such that the eigenvalues of the product matrix $DB$ are $$ 2\sin\left(\frac{k\pi}{2n+1}\right), \,\, k=1,2,\dots,n. $$ The pertaining old question: Eigenvalues of a tridiagonal trigonometric matrix