"Transforms" which appear to associate one sequence of perfect squares with another.

[creigh at o2online.de]

Dear seqfans, 

FAMP should soon be able associate several thousand sequences with  
each floretion. 

Below, 3 different "tes-transforms" of the same element are given:
+ 3'i - 2'j + 'k + 3i' - 2j' + k' - 4'ii' - 3'jj' + 4'kk' - 'ij' - 'ji' 
+ 3'jk' + 3'kj' + 4e 

1tesseq: 4, 49, 676, 9409, 131044, 1825201, 25421764, 354079489, 4931691076, 68689595569, 
956722646884

1tescycseq: 4, 4, 36, 484, 6724, 93636, 1304164, 18164644, 253000836, 3523847044, 
49080857764, 683608161636, 9521433405

1tessigcycseq: 4, 49, 36, 484, 484, 6561, 6724, 91204, 93636, 1270129, 1304164, 
17690436, 18164644, 246395809, 253000836, 3431850724, 3523847044

Notice that tessigcycseq(2n) = tescycseq(n+1) 
Concerning the above sequences, see posting to 
http://mathforum.org/discuss/sci.math/m/661493/661639 
(perhaps someone here is in a position to comment on my observation).

Another example:
 - .25'i + .5'k - .25i' - .5j' + .5k' - .75'ii' + .75'jj' - .25'kk' + .
25'jk' - .5'ki' + .25'kj' + .25e 

4tesseq: 1, 3, 4, 7, 11, 18, 29, 47, 76, 123, 199, 322, 521, 843, 1364,  //Lucas
4tescycseq: 1, -1, -1, -2, -3, -5, -8, -13, -21, -34, -55, -89, -144, -233, 
-377, -610,// Fib
(initial terms disregarded in both cases)
4tessigcycseq: 1, 3, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 
0, -1, 0, 1, 0, -1,

The "sigcyc" and "cyc" symmetries are both based upon applying a
"cyclic operator": i-> j, j -> k, k -> i after each multiplication.  As 
apparent from above, something strange about these "operators" is that (at 
least under certain conditions- if not always) they appear to transform 
one sequence of perfect squares to another (note: this "transform" is not 
unique in general- unless one includes the floretion and symmetry, i.e. 
"ves", "tes" along with whatever sequence one wishes to apply the transform 
to.) 

They also appear to be able to generate sequences which repeat themselves every 24 
terms (have no idea at the moment why this is so and probably never will):
1vessigcyc(p-ev)seq: 2, 3, 2, 1, 1, 2, 2, 2, 0, 0, 1, 2, 0, 1, 0, 1, 1, 
2, 0, 2, 2, 2, 1, 2, 2, 3, 2, 1, 1, 2, 2, 2, 0, 0, 1, 2, 0, 1, 0, 1, 1, 
2, 0, 2, 
4tessigcyc(pos)seq: 2, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 
0, 0, 1, 2, 0, 0, 1, 2, 1, 0, 0, 0, 1, 0, 0, 

Finally, they apparently also preserve the "perfect square-edness" of a 
sequence whose bisection is a sequence of perfect squares:
+ 'i - 1.5'k + i' - 1.5k' - .5'ii' - 2.5'jj' + 'kk' - 'jk' - 'kj' - e 
1tesseq: -1, 4, 11, 1, 59, 484, -1009, 6241, -2761, 13924, 
4tescycseq: -4, 9, -54, 81, -324, 81, -1782, 2025, -31428, 68121, -343926, 
408321  

To sum up- I have gotten in way over my head! The figure at 
http://www.crowdog.de/13829/20001.html is -more or less- a representation 
of what the sigcyc-sequences are 
doing (making that picture led me to define sigcyc in the first 
place).   

Sincerely, 
Creighton