Quadratic Residues and the Frobenius Coin Problem

An odd prime p has (p-1)/2 quadratic residues mod p, and for relatively prime p and q there are (p-1)(q-1)/2 non-representable Frobenius numbers. We show that there is a relationship between the non-representable Frobenius numbers of p and q and the quadratic residues of p that accounts for the presence of (p-1)/2 in both expressions