Representations by

Let rQ(n) be the representation number of a nonnegative integer n by the quaternary quadratic form . We first prove the identity rQ(p2n)=rQ(p2)rQ(n)/rQ(1) for any prime p different from 13 and any positive integer n prime to p, which was conjectured in Eum et al. (2011) [2]. And, we explicitly determine a concise formula for the number rQ(n2) as well for any integer n.