A variant of the Sylvester–Kac matrix which exhibits subsets of integer squared singular values for all orders

In [T. Boros, P. Rozsa, Linear Algebra Appl., 421 (2007), 407–416.], the authors showed that for the Sylvester–Kac matrix of odd order, $2n+1$ ( $n\ge 0$), $n$ of the squared singular values were integers. We present a variation of this matrix for which analogous results are obtained in both the odd- and even-order cases, and we derive explicit formulae for computing the corresponding squared singular values. In the even-order case, we also obtain an explicit form for the determinant. Owing to its simple matrix with a subset of easily calculated singular values makes this matrix a useful test case for numerical software for computing singular values. In addition, report several interesting empirical results regarding the singular values of this variant which we obtained using Maple high precision floating-point arithmetic.