In number theory, an octahedral number is a figurate number that represents the number of spheres in an octahedron formed from close-packed spheres. The nth octahedral number ${\displaystyle O_{n}}$ can be obtained by the formula:

The first few octahedral numbers are:

The octahedral numbers have a generating function

Sir Frederick Pollock conjectured in 1850 that every positive integer is the sum of at most 7 octahedral numbers. This statement, the Pollock octahedral numbers conjecture, has been proven true for all but finitely many numbers.

In chemistry, octahedral numbers may be used to describe the numbers of atoms in octahedral clusters; in this context they are called magic numbers.

An octahedral packing of spheres may be partitioned into two square pyramids, one upside-down underneath the other, by splitting it along a square cross-section. Therefore, the ${\displaystyle n}$th octahedral number ${\displaystyle O_{n}}$ can be obtained by adding two consecutive square pyramidal numbers together:

If ${\displaystyle O_{n}}$ is the ${\displaystyle n}$th octahedral number and ${\displaystyle T_{n}}$ is the ${\displaystyle n}$th tetrahedral number then

This represents the geometric fact that gluing a tetrahedron onto each of four non-adjacent faces of an octahedron produces a tetrahedron of twice the size.