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

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

Quick Facts Total no. of terms, Subsequence of ...

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

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The first such numbers are

• 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..

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