Square pyramidal number

A pyramidal number, or square pyramidal number, is a figurate number that represents a pyramid with a base and four sides. The nth pyramidal number is

$\sum _{k=1}^{n}k^{2}={1 \over 6}n(n+1)(2n+1)$

The first few pyramidal numbers are

1, 5, 14, 30, 55, 91, 140, 204, 285, 385, 506, 650, 819

Pyramidal numbers can be modelled in physical space with a given number of balls and a square frame that hold in place the number of balls forming the base, that is, n². Besides 1, there is only one other number that is both a square and a pyramidal number, 4900. This fact was proven by G.N. Watson in 1918.

The sum of two consecutive square pyramidal numbers is an octahedral number.

See also: tetrahedral number