Pyramidal number

File:Square pyramidal number.svg

Geometric representation of the square pyramidal number Template:Nowrap

A pyramidal number is the number of points in a pyramid with a polygonal base and triangular sides. 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. 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

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

where Template:Math, Template:Math.

This formula can be factored:

$P_n^r=\frac{n(n+1)(n(r-2)-(r-5))}{(2)(3)}=\left(\frac{n(n+1)}{2}\right)\left(\frac{n(r-2)-(r-5)}{3}\right)=T_n\cdot \frac{n(r-2)-(r-5)}{3},$

where Template:Mvar is the Template:Mvarth triangular number.

Sequences

The first few triangular pyramidal numbers (equivalently, tetrahedral numbers) are:

1, 4, 10, 20, 35, 56, 84, 120, 165, 220, ... Template:OEIS

The first few square pyramidal numbers are:

1, 5, 14, 30, 55, 91, 140, 204, 285, 385, 506, 650, 819, ... Template:OEIS.

The first few pentagonal pyramidal numbers are:

1, 6, 18, 40, 75, 126, 196, 288, 405, 550, 726, 936, 1183, 1470, 1800, 2176, 2601, 3078, 3610, 4200, 4851, 5566, 6348, 7200, 8125, 9126 Template:OEIS.

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:

1, 8, 26, 60, 115, 196, 308, 456, 645, 880, 1166, 1508, 1911, ... Template:OEIS