On the permutations that strongly avoid the pattern 312 or 231

In 2019, B\'ona and Smith introduced the notion of strong pattern avoidance, that is, a permutation and its square both avoid a given pattern. In this paper, we enumerate the set of permutations which not only strongly avoid the pattern 312 or 231 but also avoid the pattern, for S₃ and some S₄. One of them is to give a positive answer to a conjecture of Archer and Geary.