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 n {\displaystyle n} is given by ( 2 n + 1 ) ( 5 n 2 + 5 n + 3 ) 3 . {\displaystyle {\frac {(2n+1)\left(5n^{2}+5n+3\right)}{3}}.}
The first such numbers are
References Sloane, N. J. A. (ed.). "Sequence A005902 (Centered icosahedral (or cuboctahedral) numbers, also crystal ball sequence for f.c.c. lattice)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation..