Octahedral number

can be obtained by the formula:[1] 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.

can be obtained by adding two consecutive square pyramidal numbers together:[1] If

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.

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): If two tetrahedra are attached to opposite faces of an octahedron, the result is a rhombohedron.

Descartes introduced the study of figurate numbers based on the Platonic solids and some of the semiregular polyhedra; his work included the octahedral numbers.

However, De solidorum elementis was lost, and not rediscovered until 1860.

In the meantime, octahedral numbers had been studied again by other mathematicians, including Friedrich Wilhelm Marpurg in 1774, Georg Simon Klügel in 1808, and Sir Frederick Pollock in 1850.