Triangular and tetrahedral number differences of sumset sizes in additive number theory

The study of sums of finite sets of integers has mostly concentrated on sets with very small sumsets (Freiman's theorem and related work) and on sets with very large sumsets (Sidon sets and $B_h$-sets). This paper considers the full range of sumset sizes of finite sets of integers and an unexpected pattern (related to the triangular and tetrahedral numbers) that appears in the distribution of popular sumset sizes of sets of size 4.