Polygonal Numbers and The Algebra of Sequences

This edition of Geometer's Corner develops one of my all-time favorite topics to teach. What follows can be incorporated in a class that includes work on sequences and series or in a geometry class that studies polygons, where the inclusion of some algebra is appropriate. The material focuses on what are known as the Figurative or Polynomial Numbers and considers three different ways to define them. These are sequences that can be generated by diagrams like the ones shown in Figure 1.