Pyramidal number

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.[1] 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.[2] It is possible to extend the pyramidal numbers to higher dimensions.

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

Formula

The formula for the nth r-gonal pyramidal number is

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

where r ∈ ℕ, r ≥ 3. [1]

This formula can be factored:

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

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)

References

1. Weisstein, Eric W. "Pyramidal Number". MathWorld.
2. Sloane, N. J. A. (ed.). "Sequence A002414". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.