Pyramidal number
A pyramidal number is the number of points in a pyramid with a polygonal base and triangular sides.<ref name=":0" /> The term often refers to square pyramidal numbers, which have a square base with four sides, but it can also refer to a pyramid with any number of sides.<ref>Template:Cite OEIS</ref> The numbers of points in the base and in layers parallel to the base are given by polygonal numbers of the given number of sides, while the numbers of points in each triangular side is given by a triangular number. It is possible to extend the pyramidal numbers to higher dimensions.
Formula
The formula for the Template:Mvarth Template:Mvar-gonal pyramidal number is
- <math>P_n^r= \frac{3n^2 + n^3(r-2) - n(r-5)}{6},</math>
where <math>r \isin \mathbb{N}</math>, Template:Math.<ref name=":0">Template:MathWorld</ref>
This formula can be factored:
- <math>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 \cdot \frac{n(r-2)-(r-5)}{3},</math>
where Template:Mvar is the Template:Mvarth triangular number.
Sequences
The first few triangular pyramidal numbers (equivalently, tetrahedral numbers) are:
The first few square pyramidal numbers are:
The first few pentagonal pyramidal numbers are:
The first few hexagonal pyramidal numbers are:
- Template:Num, Template:Num, Template:Num, Template:Num, Template:Num, Template:Num, Template:Num, 372, 525, 715, 946, 1222, 1547, 1925 Template:OEIS.
The first few heptagonal pyramidal numbers are:<ref name="b">Template:Citation.</ref>