Pyramidal number

A pyramidal number is a figurate number that represents a pyramid with a polygonal base and a given number of triangular sides. The term is most often used to refer to square pyramidal numbers, which have a square base with four sides, but it can also refer to:

• Triangular pyramidal number (three sides)
• Square pyramidal number (four sides)
• Pentagonal pyramidal number (five sides)
• Hexagonal pyramidal number (six sides)
• Heptagonal pyramidal number (seven sides)

as well as to pyramids with higher numbers of sides

The formula for a r-gonal pyramid number is:

$P_{n}^{r}={\frac {3n^{2}+n^{3}(r-2)-n(r-5)}{6}}$

r ∈ ℕ, r ≥ 3

This formula factors to:

${\begin{aligned}P_{n}^{r}&={\frac {n(n+1)[n(r-2)-(r-5)]}{(2)(3)}}\\&=\left({\frac {n(n+1)}{2}}\right)\left({\frac {n(r-2)-(r-5)}{3}}\right)\\&=T_{n}\ \left({\frac {n(r-2)-(r-5)}{3}}\right)\\\end{aligned}}$