Possibly new sequence related to dodecahedron

[wouter meeussen]

more detail is in a 70kb Excel spreadsheet :
http://users.pandora.be/Wouter.Meeussen/DodecahedralVectorSum.xls

for 2 vertices, there are 5 (not six) different sets:

{10 pairs with norm^2 of sum = 0.000}
{30 pairs with 1.000}
{60, 2.618}
{60, 5.236}
{30, 6.854}
the norm^2 is taken with the side of the pentagons =1.
And of course 10+30+60+60+30 = 190 = 20 choose 2

For the icosahedron, I get for k=1..6
1, 3, 5, 8, 8, 12

remark that the two 'penultimate ones' (resp k=4 & 5 for icosahedral and k=8 & 9 for dodecahedral)
are equal in count. (??)

W.

----- Original Message -----
From: "N. J. A. Sloane" <njas at research.att.com>
To: <seqfan at ext.jussieu.fr>; <math-fun at mailman.xmission.com>
Cc: <wouter.meeussen at vandemoortele.com>; <wouter.meeussen at pandora.be>
Sent: Tuesday, December 28, 2004 5:26 PM
Subject: Possibly new sequence related to dodecahedron



The following is an interesting question raised by
Wouter Meeussen <wouter.meeussen at pandora.be> on Dec 27 2004:

He said:
>> Based on the symmetry point-group, how many 'different' sets of 1, 2, .., 10 vertices
>> exist on a dodecahedron?
>>
>> If I count different norms of sums of 1, 2, .., 10 unit vectors, I find:
>> 1,5,12,22,34,50,65,78,78,86
>> but there might be 'different' sets that accidentaly sum to the same resultant's norm.
>> So the given integers are a lower bound.

We can rephrase the question as follows.

Let G be the full icosahedral group, of order 120.
Let v_1, ..., v_20 be the vertices of the dodecahedron.
Let S(n) be the set of vectors

   v_{i_1} + v_{i_2} + ... + v_{i_n}

where 1 <= i_1 <= i_2 <= ... <= i_n <= 20.
Then what is s(n), the number of orbits of G on S(n)?
(so s(1) = 1, s(2) = 6, ...)

Presumably this is different from A039742:
...
%S A039742 1,6,50,475,4881,52835,593382,6849415,80757819,968400940,11773656517,
%T A039742 144791296055,1797935761182
%N A039742 Lattice animals in the fcc lattice (12 nearest
  neighbors), connected rhombic dodecahedra, edge-connected cubes.
...


Repeat for icosahedron, etc.!

NJAS