In mathematics, a pyramidal number, or square pyramidal number, is a figurate number that represents a pyramid with a base and four sides. Two equivalent ways of calculating the n-th pyramidal number are

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

that is, by adding up the squares of the first n integers, or by multiplying the nth pronic number by the nth odd number. By mathematical induction it is possible to derive one formula from the other.

The first few pyramidal numbers are:

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

.

The pyramidal numbers can also be expressed as sums of binomial coefficients thus:

$\{\{n\; +\; 2\}\; \backslash choose\; 3\}\; +\; \{\{n\; +\; 1\}\; \backslash choose\; 3\}$

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, the 70th square number and the 24th square pyramidal number. 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

Figurate numbers

Pyramidenzahl | Nombre pyramidal carré | Kvadratno piramidno število