More thoughts on floretion algebra program

[creigh at o2online.de]

Hello, 

This is my guess:
If we could get the FAMP program to run "in reverse", we would have 
an extremely handy, perhaps even unique kind of tool at our 
disposal. 

By this is meant: 
One enters in a sequence, say the Fibonacci numbers. Then 
"FAMP-Superieur" would search for four sets 
JES(Fib)  = {j1, j2, j3, ....} 
LES(Fib) = {l1, l2, l3, ....} 
TES(Fib) = {t1, t2, t3, ....} 
VES(Fib) = {v1, v2, v3, ....}   subset floretion (real) algebra

where
 jes( (j1)^n ) = jes( (j2)^n ) = jes( (j3)^n ) = ...  = Fib(n)   
 les( (l1)^n ) = les( (l2)^n ) = les( (l3)^n ) = ...  = Fib(n)
 tes( (t1)^n ) = tes( (t2)^n ) = tes( (t3)^n ) = ... = Fib(n)   
 les( (t1)^n ) = les( (t2)^n ) = les( (t3)^n ) = ...  = Fib(n)

Specifically, this would mean solving a system a polynomial 
equations. My guess at a brute force method- i.e. just plugging in 
numbers and hoping these lead to the sequence at hand, is on the order
of magnitude of 60^(16) checks for "interesting" sequences; this is a 
number which could be either greatly reduced or increased for any 
number of reasons. One also has the option of a "smart bruce 
force" method which only checks those elements which
have some sort of symmetric representation on either 
Floret's cube or star (basically, this is the only course I've 
followed to now).

For a variety of reasons, it would be nice to know if the above sets were
"closed" in some sense. Example: 

Let sig(x) be the swap operator, which reverses all "arrows" of a 
floretion (if  x = i' + 'j,  then  sig(x) = 'i + j' )   

Fib(n) = ves( (Ex)^(n+1) ) = ves( (E(sig(x)))^(n+1) ) 
E = ( - i' - i' - 'ii' + 'jj' + 'kk' + 'jk' + kj' - 1 )/4
x = 'i + 'i + 'ji' + 'ki' + 1 
sig(x) = i' + i' + 'ij' + 'ik' + 1 
[ Note: sig(E) = E ]
It follows that Ex and E(sig(x)) in VES(Fib). 
Then surely (Ex)E(sig(x)) should also have something
to do with the Fibonacci numbers. Indeed, we have
ves( ( (Ex)E(sig(x)) )^(n+1) ) = ves((ExEx)^(n+1)) 
 = ves((Ex)^(2n+2)) = = Fib(2n+1) 

In addition, 
ves( ( Ex + E(sig(x)) )^n ) = ves( ( E(x + sig(x)) )^n ) = 
= http://www.research.att.com/projects/OEIS?Anum=A077020
(disregarding signs) 

Additional commentary:
-FAMP has also led to a nonabelian, countably infinite group. 
Check out my rhyme at 
http://mathforum.org/discuss/sci.math/m/633678/634274

-The program was also updated yesterday to include the 
swap operator "sig", discussed above. In addition, 
A055997(n+1) + A053141(n+1) = A001541(n+1) + A001109(n+1) 
was also found. 

Sincerely, 
Creighton