A closed-form genus formula for three-generator numerical semigroups

Let ⟨a, b, c⟩ denote the numerical semigroup minimallygenerated by pairwise coprime positive integers a < b < c. We con-jecture a closed form formula for the genus G(⟨a, b, c⟩) the numberof positive integers not representable as non negative integer lin-ear combinations of a, b, and c valid for generators satisfying anexplicit domain condition. The formula is:G(⟨a, b, c⟩) =(a − 1)(b − 1)/2 − ϖ(c)where ϖ(c) = (b − 1) − ca − 1 is a function introduced here mea-suring the gap reduction contribution of the third generator. Thedomain condition is (a−1)(b−3) ≤ c ≤ (a−1)(b−1) with c /∈ ⟨a, b⟩,which is equivalent to 0 ≤ ϖ(c) ≤ 2. The formula has been verifiedcomputationally on over 1,259 cases with zero failures, includinggenerators up to 80 and genus values in the thousands. To theauthors’ knowledge, no closed form genus formula for the generalthree generator case has previously appeared in the literature.