On the number of representations of an integer by a linear form

Let a₁,..., a_k be positive integers generating the unit ideal, and j be a residue class modulo L = lcm(a₁,..., a _k). It is known that the function r(N) that counts solutions to the equation x₁a₁ + ... + x_ka_k = N in non-negative integers x_i is a polynomial when restricted to non-negative integers N = j (mod L). Here we give, in the case of k = 3, exact formulas for these polynomials up to the constant terms, and exact formulas including the constants for q = gcd(a₁, a₂) · gcd(a₁, a₃) · gcd(a₂, a₃) of the L residue classes. The case q = L plays a special role, and it is studied in more detail.