Octahedral number

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:[1]

${\displaystyle O_{n}=P_{n-1}+P_{n}.}$

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

${\displaystyle O_{n}+4T_{n-1}=T_{2n-1}.}$

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.

Another relation between octahedral numbers and tetrahedral numbers is also possible, based on the fact that an octahedron may be divided into four tetrahedra each having two adjacent original faces (or alternatively, based on the fact that each square pyramidal number is the sum of two tetrahedral numbers):

${\displaystyle O_{n}=T_{n}+2T_{n-1}+T_{n-2}.}$

If two tetrahedra are attached to opposite faces of an octahedron, the result is a rhombohedron.[6] The number of close-packed spheres in the rhombohedron is a cube, justifying the equation

${\displaystyle O_{n}+2T_{n-1}=n^{3}.}$

Square pyramids in which each layer has a centered square number of cubes. The total number of cubes in each pyramid is an octahedral number.

The difference between two consecutive octahedral numbers is a centered square number:[1]

${\displaystyle O_{n}-O_{n-1}=C_{4,n}=n^{2}+(n-1)^{2}.}$

Therefore, an octahedral number also represents the number of points in a square pyramid formed by stacking centered squares; for this reason, in his book Arithmeticorum libri duo (1575), Francesco Maurolico called these numbers "pyramides quadratae secundae".[7]

The number of cubes in an octahedron formed by stacking centered squares is a centered octahedral number, the sum of two consecutive octahedral numbers. These numbers are

1, 7, 25, 63, 129, 231, 377, 575, 833, 1159, 1561, 2047, 2625, ... (sequence A001845 in the OEIS)

given by the formula

${\displaystyle O_{n}+O_{n-1}={\frac {(2n-1)(2n^{2}-2n+3)}{3}}}$ for n = 1, 2, 3, ...