Let

be positive integers generating the unit
ideal, and

be a residue class modulo

. It is known that the function

that
counts solutions to the equation

in
non-negative integers

is a polynomial when restricted to
non-negative integers

. Here we give, in the
case of

, exact formulas for these polynomials up to the
constant terms, and exact formulas including the constants for

of
the

residue classes. The case

plays a special
role, and it is studied in more detail.