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 usually refers 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 an r-gonal pyramidal number is:

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

with r ∈ ℕ, r ≥ 3.

This formula can be factorized as follows:

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