A figurate number which is the sum of two consecutive pyramidal numbers, O_n = P_(n - 1) + P_n = 1/3 n(2n^2 + 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 (x(x + 1)^2)/(x - 1)^4 = x + 6x^2 + 19x^3 + 44x^4 + .... Pollock conjectured that every number is the sum of at most 7 octahedral numbers. A related set of numbers is the number of cubes in the Haűy construction of the octahedron.

Haűy construction | octahedron | truncated octahedral number