Counting numerical semigroups by genus and even gaps

Let ng be the number of numerical semigroups of genus g. We present an approach to compute ng by using even gaps, and the question: Is it true that ng+1>ng? is investigated. Let Nγ(g) be the number of numerical semigroups of genus g whose number of even gaps equals γ. We show that Nγ(g)=Nγ(3γ) for γ≤⌊g∕3⌋ and Nγ(g)=0 for γ>⌊2g∕3⌋; thus the question above is true provided that Nγ(g+1)>Nγ(g) for γ=⌊g∕3⌋+1,…,⌊2g∕3⌋. We also show that Nγ(3γ) coincides with fγ, the number introduced by Bras-Amorós (2012) in connection with semigroup-closed sets. Finally, the stronger possibility fγ∼φ2γ arises being φ=(1+5)∕2 the golden number.