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I need the formula (if there exists in closed form) for eigenvalues of the following matrix:

matrix has the size $N\times N$, on the main diagonal is constant integer entry $n<N$, then the one diagonal up and down is constant integer entry $n-1$ and so on until there will be two diagonals (up and down) with integer $1$ entry. The rest of the matrix are zeros.

So such matrix is symmetric and Toeplitz (constant on diagonals) and banded. For example if $n=2$, we have on the main diagonal $2$, the next diagonal up and down has entry $1$ and the rest of the matrix zeros. For greater n construction of the matrix is by analogy.

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