Consider the sequence V(k, n) constructed in a greedy fashion by setting a1 = k, a2 = n and defining am+1 as the smallest integer larger than am that can be written as a sum of two (not necessarily distinct) earlier terms in exactly one way\; the sequence V(2, 3), for example, is given by V(2, 3) = 2, 3, 4, 5, 9, 10, 11, 16, 22, …. If n > 3 is odd, then the sequence V(2, n) has exactly two even terms 2 and 2n if and only if n -1 is not a power of 2. In that case, V(2, n) eventually becomes a finite union of arithmetic progressions.This begs several questions. For which other initial values (k, n) will V(k, n) becomea finite union of arithmetic progressions? What happens for other initial conditions? We will discuss several families of sequences defined by greedy algorithm and additive conditions similar to those defining V(k,n). These sequences have a very rich structure, much of which has yet to be understood. This is related to Sierpiński Triangle and the classical family of sequences defined by Ulam