Elements with unique length factorization of a numerical semigroup generated by three consecutive numbers

Let $S$ be the numerical semigroup generated by three consecutive numbers $a,a+1,a+2$, where $a\in\mathbb{N}$, $a\geq 3$. We describe the elements of $S$ whose factorizations have all the same length, as well as the set of factorizations of each of these elements. We give natural partitions of this subset of $S$ in terms of the length and the denumerant. By using Ap\'ery sets and Betti elements we are able to extend some results, first obtained by elementary means.