Factorization and catenary degree in 3-generated numerical semigroups

Given a numerical semigroup S ( A ) , generated by A = { a , b , N } ⊂ N with 0 < a < b < N and gcd ( a , b , N ) = 1 , we give a parameterization of the set F ( m ; A ) = { ( x , y , z ) ∈ N 3 | x a + y b + z N = m } for any m ∈ S ( A ) . We also give the catenary degree of S ( A ) , c ( A ) . Boths results need the computation of an L-shaped tile, related to the set A, that has time-complexity O ( log N ) in the worst case.