centered icosahedral number

{{Short description|Three-dimensional figurate centered icosahedral numbers}}

{{Infobox integer sequence

| number = Infinity

| parentsequence = Polyhedral numbers

| formula = $\backslash frac\{(2n+1)\backslash ,(5n^2+5n+3)\}\{3\}$

| first_terms = 1, 13, 55, 147, 309, 561, 923

| OEIS = A005902

| OEIS_name = Centered icosahedral

}}

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 $n$ is given by $$\backslash frac\{(2n+1)\backslash left(5n^2+5n+3\backslash right)\}\{3\}.$$

The first such numbers are

{{bi|left=1.6|1, 13, 55, 147, 309, 561, 923, 1415, 2057, 2869, 3871, 5083, 6525, 8217, ... {{OEIS|A005902}}.}}