Polygonal Number

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 \(c\) sides and if, on each side, there are \(n\) points arranged, vertices included, then we distinguish two kinds of polygonal numbers:

  1. Those that are represented on the perimeter of the polygon: \(\textrm{p}_{n}^{c}\) = \(c\)(\(n\) – 1)
  2. Those that are represented on the closed surface of the polygon: \(\textrm{P}_{n}^{c}\) = \(\dfrac{(c\space –\space 2)n^2 –\space (c\space –\space 4)n}{2}\)


For triangular numbers :

  1. \(\textrm{p}_{n}^{3}\) = 3(\(n\) – 1)   et   \(\textrm{p}_{5}^{3}\) = 3(5 – 1) = 12
  2. \(\textrm{P}_{n}^{c}\) = \(\dfrac{(c\space –\space 2)n^2 –\space (c\space –\space 4)n}{2}\)   et   \(\textrm{P}_{n}^{3}\) = \(\dfrac{n(n + 1)}{2}\)  et  \(\textrm{P}_{5}^{3}\) = \(\dfrac{5(5 + 1)}{2}\) = 15

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