Canonical trace ideal and residue for numerical semigroup rings

For a numerical semigroup ring K[H] we study the trace of its canonical ideal. The colength this ideal is called residue H. This invariant measures how far H from being symmetric, i.e. Gorenstein ring. We remark that contains conductor ideal, and bounds for residue. 3-generated semigroups give explicit formulas Thus, in setting can classify those whose at most one (the nearly ones), show eventual periodic behaviour shifted family.