• Source: 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



    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..

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