Counting numerical semigroups by genus and even gaps

Let $n_g$ be the number of numerical semigroups of genus $g$. We present an approach to compute $n_g$ by using even gaps, and the question: Is it true that $n_{g+1}>n_g$? is investigated. Let $N_{\gamma }(g)$ be the number of numerical semigroups of genus $g$ whose number of even gaps equals $\gamma$. We show that $N_{\gamma }(g)=N_{\gamma }(3\gamma )$ for $\gamma \le \lfloor g\mathrm{/}3\rfloor$ and $N_{\gamma }(g)=0$ for $\gamma >\lfloor 2g\mathrm{/}3\rfloor$; thus the question above is true provided that $N_{\gamma }(g+1)>N_{\gamma }(g)$ for $\gamma = \lfloor g/3 \rfloor +1, \ldots, \lfloor 2g/3\rfloor$\mathrm{.}Wealsoshowthat$N_\gamma(3\gamma)$coincideswith$f_\gamma$, the number introduced by Bras-Amor'os in conection with semigroup-closed sets. Finally, the stronger possibility $f_\gamma \sim \varphi^{2\gamma}$arisesbeing$\varphi = (1+\sqrt{5})/2$ the golden number.