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 ^[1]

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:

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

References

1. http://oeis.org/A002414