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 $n$ th $r$ -gonal pyramidal number is

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

where $r \in ℕ$ , $r \geq 3$ . [1]

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 $n$ th triangular number.

References

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

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[:0-1] Weisstein, Eric W. "Pyramidal Number". MathWorld.

[2] Sloane, N. J. A. (ed.). "Sequence A002414". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.

Pyramidal number

Formula

See also

References