The Betti numbers and Golodness of numerical semigroup rings of Sally type

We determine explicit formulae for the Betti numbers of the defining toric ideals of two families of numerical semigroups of Sally type, resolving and extending beyond several conjectures of Goel–Şahin–Singh–Srinivasan. Our approach uses Apéry resolutions and their specialisations. For the family $${\mathrm{\Gamma }}_m(n)=⟨m,m+1,\dots ,\widehat{m+n},\dots ,2m-1⟩,2\le n<m,$$ where the hat denotes omission, we also compute the Poincaré series of the residue field and characterise Golodness. For fixed multiplicity $m$ and $3\le n<m$, we show that these rings share the same rational residue field Poincaré series, but exhibit arbitrarily late departures from Serre’s upper bound.