Even more terms for A108973 and A124505...

[Andrew Weimholt]

I have submitted two more terms to A108973, the number of regular 
(4n+3)-simplices
that can be inscribed on a (4n+3)-cube...

          8898131405512141870083342336000 = 2^41 * 3^10 * 5^3 * 7^3 * 11 * 
13 * 17 * 19 * 34603
10827543712227210782977570287648768000000 = 2^52 * 3^9 * 5^6 * 7^4 * 11^2 * 
13 * 17 * 19 * 23 * 278617

The large prime factors at the end can be found in A048615, which is no 
coincidence, since
the problem of finding simplices on the cube is related to Hadamard 
matrices.

%I A108973
%S A108973 2,480,10321920,64561751654400,1480206036768915456000,
8898131405512141870083342336000,10827543712227210782977570287648768000000
%O A108973 1
%K A108973 ,more,nonn,
%A A108973 Andrew Weimholt (andrew at weimholt.com), Dec 26 2006
RH
RA 192.20.225.32
RU
RI

(The offset should be 0 - I missed correcting the sample data on the form)

I also updated the alternate version of this sequence, A124505, the number 
of regular n-simplices
that can be inscribed on a n-cube...

%I A124505
%S A124505 1, 1, 0, 2, 0, 0, 0, 480, 0, 0, 0, 10321920, 0, 0, 0, 
64561751654400, 0, 0, 0, 1480206036768915456000, 0, 0, 0, 
8898131405512141870083342336000, 0, 0, 0, 
10827543712227210782977570287648768000000, 0, 0, 0
%O A124505 0
%K A124505 ,more,nonn,
%A A124505 Andrew Weimholt (andrew at weimholt.com), Dec 26 2006
RH
RA 192.20.225.32
RU
RI

(For this one, the second offset is incorrect (is 1 should be 4), but I do 
not know how to correct it).

These additional terms were computed using the orders of the automorphism 
groups
of Hadamard matrices which can be found at 
http://www.research.att.com/~njas/hadamard/index.html

Up until last week, I didn't know that this data was so readily available, 
and to be honest,
up until a couple months ago, I hadn't heard of "Hadamard", so the previous 
terms of the sequence
were computed with much more effort than these last two.

The number of regular (4n-1)-simplices which can be inscribed on an 
(4n-1)-cube is precisely the number of
regular 4n-orthoplexes which can be inscribed on a 4n-cube. Each Hadamard 
matrix corresponds
to such an orthoplex, and the automorphism group of the Hadamard matrix is 
also the automorphism
group of the corresponding orthoplex.

For each "type" of orthoplex, we can compute the number appearing on the 
cube by dividing the
order of the symmetry group of the cube, by the order of the automorphism 
group of the orthoplex.
We then sum the results for each type to get the total number of 
orthoplexes on the cube.

For the A108973(4) term that I submitted last week, I commented on the 
orders of the
automorphism groups of the simplices for dimensions 19 and 15, and later 
noticed this relationship...

 Let H be the automorphism group for a Hadamard matrix of order 4n
 Let G be the automorphism group of a corresponding simplex on cube 
(dimension 4n-1)

 for (n<6)

   8n * |Aut(G)| = |Aut(H)|

 This holds for n<6 because the all of the 8n cubically aligned (4n-1)-s  
implices of the
 4n-orthoplexes are equivalent. However, for n>6, some of the orthoplexes 
have multiple
 types of cubically aligned simplices. (By "cubically aligned", I mean 
those (4n-1)-simplices
 of the 4n-orthoplex which are inscribed on (4n-1)-cubes of the 4n-cube.)

  In short, for n>6, there is no >one< "corresponding simplex", but a set 
of inequivalent simplices.

The number of inequivalent 4n-orthoplexes on the 4n-cube is the same as the 
number of
inequivalent Hadamard matrices of order 4n (see A007299). For n<6, this is 
also the number
of inequivalent (4n-1)-simplices on the (4n-1)-cube, but for n=6, the 
number of inequivalent
simplices is at least 110 (it'll take some time to determine the precise 
number).

Andrew