Parity Difference in the Frobenius Coin Problem - One Odd and Two Even Parameters

This paper studies the parity distribution of nonrepresentable integers in a generalized Frobenius coin problem. Let a be an odd positive integer and b,c be even positive integers satisfying gcd⁡(a,b,c)=1. Define T as the set of positive integers that cannot be represented in the form ax+by+cz with x,y,z≥0. Through computational experiments and a proof based on a residue pipe model, we show that the difference between the number of odd and even elements of T is always (a−1)/2. This result generalizes the parity phenomenon appearing in PROMYS 2025 Problem 6 and establishes that the parity difference depends only on the odd parameter a.