Polygonal Number
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 [latex]c[/latex] sides and if, on each side, there are [latex]n[/latex] points arranged, vertices included, then we distinguish two kinds of polygonal numbers:
- Those that are represented on the perimeter of the polygon: [latex]\textrm{p}_{n}^{c}[/latex] = [latex]c[/latex]([latex]n[/latex] – 1)
- Those that are represented on the closed surface of the polygon: [latex]\textrm{P}_{n}^{c}[/latex] = [latex]\dfrac{(c\space –\space 2)n^2 –\space (c\space –\space 4)n}{2}[/latex]
Example
For triangular numbers :- [latex]\textrm{p}_{n}^{3}[/latex] = 3([latex]n[/latex] – 1) et [latex]\textrm{p}_{5}^{3}[/latex] = 3(5 – 1) = 12
- [latex]\textrm{P}_{n}^{c}[/latex] = [latex]\dfrac{(c\space –\space 2)n^2 –\space (c\space –\space 4)n}{2}[/latex] et [latex]\textrm{P}_{n}^{3}[/latex] = [latex]\dfrac{n(n + 1)}{2}[/latex] et [latex]\textrm{P}_{5}^{3}[/latex] = [latex]\dfrac{5(5 + 1)}{2}[/latex] = 15