Pyramidal number

Geometric representation of the square pyramidal number 1 + 4 + 9 + 16 = 30.

A pyramidal number is a figurate number that represents a pyramid with a polygonal base and a given number of triangular sides. A pyramidal number is the number of points in a pyramid where each layer of the pyramid is an r-sided polygon of points. The term usually refers to square pyramidal numbers, which have a square base with four sides, but it can also refer to pyramids with three or more sides. It is possible to extend the pyramidal numbers to higher dimensions.

Formula

The formula for the nth r-gonal pyramidal number is

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

where r ∈ ℕ, r ≥ 3.

This formula can be factored:

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

where T_n is the nth triangular number.

See also

• 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)