Centered octahedral number

A centered octahedral number or Haüy octahedral number is a figurate number that counts the number of points of a three-dimensional integer lattice that lie inside an octahedron centered at the origin. The same numbers are special cases of the Delannoy numbers, which count certain two-dimensional lattice paths. The Haüy octahedral numbers are named after René Just Haüy.

Centered octahedral number

Haüy construction of an octahedron by 129 cubes
Named after	René Just Haüy
Publication year	1801
Total no. of terms	Infinity
Subsequence of	Polyhedral numbers, Delannoy numbers
Formula	${\displaystyle {\frac {(2n+1)\left(2n^{2}+2n+3\right)}{3}}}$
First terms	1, 7, 25, 63, 129, 231, 377
OEIS index	A001845 Centered octahedral