Total no. of terms Infinity Polyhedral numbers ${\displaystyle {\frac {(2n+1)\,(5n^{2}+5n+3)}{3))}$ 1, 13, 55, 147, 309, 561, 923 A005902Centered icosahedral

The centered icosahedral numbers and cuboctahedral numbers are two different names for the same 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

1, 13, 55, 147, 309, 561, 923, 1415, 2057, 2869, 3871, 5083, 6525, 8217, ... (sequence A005902 in the OEIS).

## 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..