A floretion "transform" of the helious sequence

[creigh at o2online.de]

Without going into any detail at the moment (I would like to wait a little 
and see for myself exactly what it is I'm doing, first) it seems possible 
to "force" the sequence "vesseq" to take on any values one wishes. Basically, 
the method consists of altering the floretion itself so that it 
takes on the values of the sequence (or lax but better: one "asks" a floretion 
to take on certain values - and it probably will...  on its "own conditions") 
.   In my opinion, this or some variation of it could eventually be of upmost 
interest because it allows one to create many  (at the moment on the order 
of 1-1000000) different types of transforms of any sequence one wishes (with 
relationships among the sequences provided automatically by the various 
identities such as ves = tes + les + jes, etc  for -almost- any such transform)  

To start, I "force" a certain floretion (whose "ves" would normally take on 
Fibonacci numbers, A000045) to take on my "helious sequence", instead. Here 
is what happens:

1vesforcycseq: 0, 2, 1, 3, 3, 4, 3, 6, 3, 6, 3, 7, 5, 8, 5, 11, 3, 6, 3, 
9, 9, 12, 9, 16, 5, 10, 5, 13, 11, 16, 11, 22, 3, 6, 3, 9, 9, 12, 9, 18, 
9, 18, 9, 21, 15, 24, 15, 31, 5, 10, 5, 15, 15, 20, 15, 28, 11, 22, 11, 
27, 21, 32, 21, 43, 3, 6, 3, 9, 9, 12, 9, 18, 9, 18, 9, 21, 15, 24, 15, 
32, 8, 16, 8, 32, 32, 928,  
 
 4tesforcycseq: 1, 15, 11, 20, 25, 34, 40, 67, 87, 142, 214, 357, 570, 929, 
1491, 2436, 3913, 6346, 10282, 16685, 27024, 43751, 70793, 114590, 185388, 
300009, 485461, 785578, 1271145, 2056822, 3328040, 5384979, 8713063, 1409811, 
22811334, 36909661, 59721212,  
 
 4lesforcycseq: -1, -3, -5, -10, -17, -28, -44, -67, -105, -160, -250, -389, 
-616, -977, -1561, -2498, -4017, -6460, -10414, -16805, -27150, -43883, 
-70955, -114748, -185608, -300241, -485723, -785828, -1271413, -2057096, 
-3328356, -5385283, -8713465, -14098544,  
 
 2jesforcycseq: 0, -2, -1, 1, 2, 5, 8, 12, 15, 21, 24, 30, 33, 40, 45, 53, 
58, 69, 72, 78, 81, 90, 99, 111, 120, 136, 141, 151, 156, 169, 180, 196, 
207, 229, 232, 238, 241, 250, 259, 271, 280, 298, 307, 325, 334, 355, 370, 
394, 409, 440, 445, 455, 460, 475, 490, 510, 525, 553, 564, 586, 597, 

Notice that, via the trusty identity "ves = jes + les + tes", the helious sequence 
has been split into the sum of three strictly increasing/decreasing sequences 
(disregarding initial terms) and that the sequence of ratios from "tesfor" 
and "lesfor" appear to go to the golden ratio. Note: I would have really 
liked to have put an explanation point behind that last sentence, but with 
4 hours sleep (of the last 48), I know I could again very well be overlooking 
something highly trivial.   

What about the last one, "jesfor"? Using jesleft + jesright = jes, we can 
split it into two sequences:  

4jesleftforcycseq: 1, -2, 4, 8, 16, 27, 43, 64, 96, 142, 211, 317, 484, 750, 
1174, 1852, 2944, 4699, 7525, 12092, 19479, 31430, 50759, 82021, 132595, 
214406, 346760, 560904, 907390,  
 
 4jesrightforcycseq: -1, -2, -6, -6, -12, -17, -27, -40, -66, -100, -163, -257, 
-418, -670, -1084, -1746, -2828, -4561, -7381, -11936, -19317, -31250, -50561, 
-81799, -132355, -214134, -346478,  

which again appear to be increasing/decreasing (and where the sequence of 
ratios again appears to approach the golden ratio).

Here's another where I "force a ves" (corresponding to a certain floretion) to 
take on Fibonacci numbers;

1vesforseq: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 
1597, 2584, 4181, 
 
 4tesforseq: -5, 9, -1, 13, 12, 30, 47, 87, 149, 261, 450, 776, 1331, 2277, 
3883, 6605, 11208, 18978, 32071, 54099, 91105, 153189, 257214, 431308, 722347, 
1208385, 2019287, 3370957,  
 
 4lesforseq: 5, -3, 7, 1, 10, 10, 21, 31, 53, 85, 140, 228, 373, 609, 995, 
1625, 2654, 4334, 7077, 11555, 18865, 30797, 50272, 82056, 133925, 218565, 
356671, 582001, 949618, 1549330,  
 
 2jesforseq: 0, -1, -1, -3, -5, -10, -18, -33, -59, -105, -185, -324, -564, 
-977, -1685, -2895, -4957, -8462, -14406, -24465, -41455, -70101, -118321, 
-199368, -335400, -563425, -945193, 

Having a quick glance, jesfor, it apparently corresponds to
http://www.research.att.com/projects/OEIS?Anum=A010049
This is a sequence I have not seen come up in my studies until now.

Sincerely, 
Creighton