In mathematics, the centered icosahedral numbers also known as cuboctahedral numbers are a sequence of numbers, describing two different representations for these numbers as three-dimensional figurate numbers. As centered icosahedral numbers, they are centered numbers representing points arranged in the shape of a regular icosahedron. As cuboctahedral numbers, they represent points arranged in the shape of a cuboctahedron, and are a magic number for the face-centered cubic lattice. The centered icosahedral number for a specific ${\displaystyle n}$ is given by $${\displaystyle {\frac {(2n+1)\left(5n^{2}+5n+3\right)}{3}}.}$$

The first such numbers are