More Brieskorn spheres bounding rational balls

We call an integral homology sphere $\mathit{\text{non-trivially}}$ bounds a rational homology ball if it is obstructed from bounding an integral homology ball. After Fintushel and Stern’s well-known example $\mathrm{\Sigma }(2,3,7)$, Akbulut and Larson recently provided the first infinite families of Brieskorn spheres non-trivially bounding rational homology balls: $\mathrm{\Sigma }(2,4n+1,12n+5)$ and $\mathrm{\Sigma }(3,3n+1,12n+5)$ for odd~$n$. Using their technique, we present new such families: $\mathrm{\Sigma }(2,4n+3,12n+7)$ and $\mathrm{\Sigma }(3,3n+2,12n+7)$ for even $n$. Also manipulating their main argument, we simply recover some classical results of Akbulut and Kirby, Fickle, Casson and Harer, and Stern about Brieskorn spheres bounding integral homology balls.