Editing Octahedral number (section)

==Relation to other figurate numbers==

===Square pyramids===
An octahedral packing of spheres may be partitioned into two [[square pyramid]]s, one upside-down underneath the other, by splitting it along a square cross-section. Therefore,
the {{nowrap|<math>n</math>th}} octahedral number <math>O_n</math> can be obtained by adding two consecutive [[square pyramidal number]]s together:<ref name="bon"/>
:<math>O_n = P_{n-1} + P_n.</math>

===Tetrahedra===
If <math>O_n</math> is the {{nowrap|<math>n</math>th}} octahedral number and <math>T_n</math> is the {{nowrap|<math>n</math>th}} [[tetrahedral number]] then
:<math>O_n+4T_{n-1}=T_{2n-1}.</math>
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):
:<math>O_n = T_n + 2T_{n-1} + T_{n-2}.</math>

===Cubes===
If two tetrahedra are attached to opposite faces of an octahedron, the result is a [[rhombohedron]].<ref>{{citation|first=John G.|last=Burke|title=Origins of the science of crystals|publisher=University of California Press|year=1966|page=88|url=https://books.google.com/books?id=qvxPbZtJu8QC&pg=PA88}}.</ref> The number of close-packed spheres in the rhombohedron is a [[Cube (algebra)|cube]], justifying the equation
:<math>O_n+2T_{n-1}=n^3.</math>

===Centered squares===
[[File:Pyramides quadratae secundae.svg|thumb|upright=1.35|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]]:<ref name="bon"/>
:<math>O_n - O_{n-1} = C_{4,n} = n^2 + (n-1)^2.</math>
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".<ref>[http://www.maurolico.unipi.it/edizioni/arithmet/ariduo/ari1/ari1-2.htm Tables of integer sequences] {{Webarchive|url=https://archive.today/20120907110907/http://www.maurolico.unipi.it/edizioni/arithmet/ariduo/ari1/ari1-2.htm |date=2012-09-07 }} from ''Arithmeticorum libri duo'', retrieved 2011-04-07.</ref>

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, ... {{OEIS|A001845}}
given by the formula 
:<math>O_n+O_{n-1}=\frac{(2n-1)(2n^2-2n+3)}{3}</math> for ''n'' = 1, 2, 3, ...