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, a≥ 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.