Kissing^3 number

[y.kohmoto]

    Hello, Seqfans.
    A news related with kissing number.
    http://www.sciencenews.org/20040214/fob7.asp
    Does anyone know the kissing number of M&Ms candy?

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    I defined K^m number of n-dimension sphere as follows.
    Where K for "Kissing".
    Let's assume "K^m" to be a verb which means "kiss m time".

    " Different two spheres X  K^ms  Y " is defined as follows.
    {(m-1) spheres X_i . i=1 to m-1 exist and     X_i kisses X_{i+1} , for
i=0 to m-1 . }

         X  - X_1 - .... - X_{m-1} - Y

         and

    Not { m' spheres X_i . i=1 to m'  exist and    X_i kisses X_{i+1} , for
i=0 to m'   } , for m' = 0 to m-2

    Not  X  - X_1 - .... - X_{m-2} - Y
               ....
    Not  X  - X_1 -  Y
    Not  X  -  Y

    Where X=X_0 and Y=X_m .

    "K^m number of n dimension sphere" is defined as "Maximal number of
spheres which K^m one sphere.". Where a sphere means  n dimension sphere..

    e.g.
    "K^3 number of 2 dimension sphere" :
    Number of circles which K^3 one circle.
    Number of  circle Ys such  that  { X - o - o - Y } but not{ X - o - Y }
not{ X - Y }.
    e.g.
    "K^1 number of n dimension sphere" :
    It is the same thing as kissing numner on n dimension.

    I would like to enumerate "K^3 number of n dimension sphere."

    %S A000001 2, 20,
    %N A000001 a(n) gives "K^3 number of n dimension sphere."

    But it is difficult.
    Please someone calculate them for n=3, 4, ....

    Comment :
    The densest packing of circles gives 18 for a(2), but it is not the
maximal number.

    Yasutoshi