Centered icosahedral number

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 $${\displaystyle {\frac {(2n+1)\left(5n^{2}+5n+3\right)}{3}}.}$$

Centered icosahedral number

Total no. of terms	Infinity
Subsequence of	Polyhedral numbers
Formula	${\displaystyle {\frac {(2n+1)\,(5n^{2}+5n+3)}{3}}}$
First terms	1, 13, 55, 147, 309, 561, 923
OEIS index	A005902 Centered icosahedral

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..