The sequence A001653

[creigh at o2online.de]

This is one of the first sequences found based upon 3-dim aspects 
of "Floret's cube" http://www.crowdog.de -> The Floretions -> 
bottom of page

Define x = .5(-'i +  i' + 'ii' - 'jj' - 'kk' + 1)

In that picture " 'i " is represented by any one of the three tips 
at the top of the "cube" [the object seems to comprise three "tilted 
pyramids" ]. The other elements making up  x, namely 
i', 'ii', 'jj', 'kk', 1 are elements which are either directly on the pyramid 
or (in the case of 'ii' and 1) symmetric to it. Writing " -'i " instead 
of " 'i " in x further "emphasizes" (in a way I cannot describe exactly) 
that 'i should be treated special (indeed, it is the tip of the pyramid)... 
so should  'jj' and 'kk' because they are on the pyramid whilst 'ii' is 
not.
 
We have ( ves( x^n ) ) = (0, -1, -2, -3, -4, -5, -6, -7, -8, )
Nothing spectacular about that sequence in itself...
unless if the number of multiplications involved
 ( = 36 for each n after simplifying ) is taken into account; 
this results in a kind of "silent beauty". 

Now, I define y = .5( 'i -  i' + 'ii' - 'jj' - 'kk' + 1). By doing so, we 
are 
adding even more "symmetry" to the problem because there is actually
no reason to favor 'i over i' when setting up the cube. Of course, 
( ves( y^n) ) =   (0, -1, -2, -3, -4, -5, -6, -7, -8, ...) also.

We have  xy = (3/2)'ii' - 'jj' - 'kk' - 'jk' + 'kj' + (3/2)1
and ves ( (xy)^n ) = (1, 5, 29, 169, 985, ...) 
which is 
http://www.research.att.com/projects/OEIS?Anum=A001653

Sincerely, 
Creighton