Octahedral Number

A figurate number which is the sum of two consecutive pyramidal numbers,

(1)

The first few are 1, 6, 19, 44, 85, 146, 231, 344, 489, 670, 891, 1156, ... (OEIS A005900). The generating function for the octahedral numbers is

(2)

Pollock (1850) conjectured that every number is the sum of at most 7 octahedral numbers (Dickson 2005, p. 23).

A related set of numbers is the number of cubes in the Haűy construction of the octahedron. Each cross section has area

(3)

where is an odd number, and adding all cross sections gives

(4)

for an odd number. Re-indexing so that gives

(5)

the first few values of which are 1, 7, 25, 63, 129, ... (OEIS A001845). These numbers have the generating function

(6)

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