That's not my area of expertise, but I wonder if this coin problem has been asked (and answered) before. I only found this similar question but it seems to me that the solution for the coins (3, 7, 9) cannot be applied to (5, 7, 11).
@MartinR I have referred askers to this answer by Robjohn whenever the question involves two coins (of coprime values). Its answer gives a useful bound for three coin problems whenever two of them are coprime, and we can apply brute force to check the finitely many missing sums below the guaranteed bound.
I dont't know if there is a comprehensive of theory for the case of three coins. The techniques become more diverse when there are more coins, and there is scope for trickery.
