  A real persymmetric Jacobi matrix of order $n$ whose eigenvalues are $2k^2$
$(k=0, ..., n-1)$ is presented, with entries given as explicit functions of
$n$. Besides the possible use for testing forward and inverse numerical
algorithms, such a matrix is especially relevant for its connection with the
dynamics of a mass-spring chain, which is a multi-purpose prototype model.
Indeed, the mode frequencies being the square roots of the eigenvalues of the
interaction matrix, one can shape the chain in such a way that its dynamics be
perfectly periodic and dispersionless.
