Possibly new sequence related to dodecahedron

[David Wilson]

Please check A030135-A030138.

At the time, I submitted sequences for each of the platonic solids.  As I
remember, the sequences for tetrahedron, cube, and octahedron were
rejected by NJAS for some reason or other.

----- Original Message ----- 
From: "Franklin T. Adams-Watters" <franktaw at netscape.net>
To: <ham>; <seqfan at ext.jussieu.fr>; <math-fun at mailman.xmission.com>
Sent: Thursday, December 30, 2004 8:16 PM
Subject: RE: Possibly new sequence related to dodecahedron


>A pair of related, also finite, sequences: the number of "polypents" 
>(connected sets of pentagons, up to symmetry) on an dodecahedron, and 
>"polyiamonds" on an icosohedron.
>
> These sequences can also be done for smaller Platonic solids, but are too 
> simple to be interesting.  This is especially so for the tetrahedron, 
> where the sequence is:
>
> 1,1,1,1,1
>
> For the square, we have:
>
> 1,1,1,2,2,1,1.
>
> For the octahedron, the sequence is (I think):
>
> 1,1,1,1,3,3,3,1,1
>
> "N. J. A. Sloane" <njas at research.att.com> wrote:
>>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
>>
>
>
> -- 
> Franklin T. Adams-Watters
> 16 W. Michigan Ave.
> Palatine, IL 60067
> 847-776-7645
>
>
> __________________________________________________________________
> Switch to Netscape Internet Service.
> As low as $9.95 a month -- Sign up today at 
> http://isp.netscape.com/register
>
> Netscape. Just the Net You Need.
>
> New! Netscape Toolbar for Internet Explorer
> Search from anywhere on the Web and block those annoying pop-ups.
> Download now at http://channels.netscape.com/ns/search/install.jsp 




MIME-Version: 1.0