Octahedral number
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 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 number is the sum of at most 7 octahedral numbers: see Pollock octahedral numbers conjecture.
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 nth octahedral number can be obtained by adding two consecutive square pyramidal numbers together:
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