Figurate number that we can represent by a convex regular polygon or by a sequence that is partially superimposed on convex regular polygons.
If the convex regular polygon has c sides and if, on each side, there are n points arranged, vertices included, then we distinguish two kinds of polygonal numbers:
- Those that are represented on the perimeter of the polygon: pcn = c(n – 1)
- Those that are represented on the closed surface of the polygon: Pcn = (c – 2)n2– (c – 4)n2
Example
For triangular numbers :
- p3n = 3(n – 1) et p35 = 3(5 – 1) = 12
- Pcn = (c – 2)n2– (c – 4)n2 et P3n = n(n+1)2 et P35 = 5(5+1)2 = 15