Strange new sequences

[Creighton Dement]

Dear Seqfans, 

I'm coming across a new class of floretion sequences which, laxly
stated, appear to go from being
I:  “disordered” to “ordered” 
II:  “disordered” to periodic (“ordered”)
III: disordered to ordered and back to disordered 

An example of Case I is given at  http://www.crowdog.de/JesBrosbEx1.html
 
There, 3 sequences are plotted together: Colors blue and red correspond
to “jesrightcycsumseq” and “jesleftcycsumseq”. The color black
corresponds to “jescycsumseq” with jescycsumseq =   2*jesrightcycsumseq
+  2*jesleftcycsumseq

As an example of Case II, I found the sequence 
http://www.crowdog.de/tescycsum3000.jpg which goes to 155[2788],
-46[2789], 52[2790], -83[2791], -154[2792], 38[2793], -55[2794],
-83[2795], 21[2796], 146[2797], -183[2798], -11[2799], 61[2800],
-147[2801], 183[2802], 10[2803], -61[2804], 146[2805], -183[2806],
-11[2807], 61[2808], -147[2809], 183[2810], 10[2811], -61[2812],
146[2813], -183[2814], -11[2815], 61[2816], -147[2817], 183[2818],
10[2819], -61[2820], 146[2821], -183[2822], -11[2823], 61[2824],
-147[2825], 183[2826], 10[2827],  20], -147[2921], 183[2922], 10[2923],
-61[2924], 146[2925], -183[2926],  

where the number in brackets gives the n-th term of the sequence and the
sequence (a(n+2797)) is (apparently- with 6000 terms checked) periodic
with period 8.  


Does anyone have suggestions, for ex., as to where I should look for
similar sequences which already exist in the OEIS?

The "quaternion loop" sequences A108618, etc. 
http://www.research.att.com/projects/OEIS?Anum=A108618
are "cousins" of the above because both rely on summing up over
fractional parts of basis vector coefficients. On the same token, I
would like to mention that the sequence 
http://www.research.att.com/projects/OEIS?Anum=A108985
appears to be very good news... it means that you can use the same rules
to generate a "crazy" sequence with no hope of finding a g.f. as well as
those with ("well-behaved") g.f.'s - whether a given sequence falls into
one category or the other will solely depend on the floretion (i.e. or
quaternion) used.

Sincerely, 
Creighton