Compositions with 3 Pairwise Coprime Parts

How many ways can we write $n$ as a sum of $3$ positive integers, no pair of which share a common factor? We express this quantity in terms of the number of solutions to a certain class of linear Diophantine equations. This allows us to show that there are $$\prod_{p \mid n} \left( 1- \frac{1}{p^2} \right) \prod_{q \nmid n} \left( 1- \frac{3}{q^2} \right) \frac{n^2}{2} + O(n^{3/2+o(1)})$$ such compositions, where the products are over primes that respectively do and don't divide $n$. This strengthens the previous result of Bubbolini, Luca, and Spiga (arXiv:1202.1670)