On force(d)-transforms

Tue Feb 15 14:26:46 CET 2005

Dear Seqfans, 

I start with the simple floretion  
 + .5j' + .5'ji' + .5'jk' + .5e 

It is has the symmetric property  j' * 'ji' * 'jk' * e = e

In August of last year, I only had jes, ves, les and tes and my disposal
- I would have checked these for the floretion and gotten:

 1vesseq: 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2,   

 2tesseq: 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,  

 1lesseq: 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 

 2jesseq: 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 

Then I would have said "this one is too simple to be of any interest"
and immediately moved on to more "complicated"  floretions (all in all,
I encountered situations like this many hundreds of times).  Now, I
attempt to show the reader that I  was being naive... 

We can apply a force-transformation to any sequence desired ( in our
scenario here, vesseq will take on this sequence; in other variants it
is tesseq which takes on the sequence ).  The "golden rule of force
transforms" ( as  described in the second paragraph at
http://www.crowdog.de/20801/21101.html  ) means that the zero-sequence 
should be transformed first. This yields: 

1vesforseq: 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,  

 2tesforseq: -3, 3, -3, 3, -3, 3, -3, 3, -3, 3,   

 1lesforseq: 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, 

 2jesforseq: 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1,  

The "golden rule" sais: do not continue tranforming any other sequences
unless these sequences are already known.   In this case it is obvious
that the sequences are well-known. But how does one "know" which
sequences to transform  (say, to get to something which is not
"trivial")? We can start with those already there! This should add an
additional  element of "symmetry" to the case at hand.  Let's start with
lesseq and keep transforming in the following loop:

Step 1: Transform lesseq by forcing vesseq to take on its values. 
Step 2: Step one will have yielded a new sequence- the sequence
"lesforseq". Repeat step one (replacing the words  "lesseq" with
"lesforseq" and "vesseq" with "vesforseq").

This leads us directly to the results, below: 

First iteration (les): 

1vesforseq: 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,  

 2tesforseq: -1, 2, -1, 2, -1, 2, -1, 2, -1, 2, -1, 2, -1,  

 1lesforseq: 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1,  

 2jesforseq: 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 

Second iteration (les):

1vesforseq: 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0,  

 2tesforseq: -1, 0, 2, -3, 5, -6, 8, -9, 11, -12, 14, -15, 17, 
Congruent to 0 or 2 mod 3

 1lesforseq: 1, 0, 0, 1, -1, 2, -2, 3, -3, 4, -4, 5, -5, 6, -6, 7, -7,
8, -8,  

 2jesforseq: 1, 0, 0, 1, -1, 2, -2, 3, -3, 4, -4, 5, -5, 6, -6, 7, -7,
8, -8,  

Third iteration (les): 

1vesforseq: 1, 0, 0, 1, -1, 2, -2, 3, -3, 4, -4, 5, -5, 6, -6, 7, -7, 8,
-8, 9, -9, 10, - 

 2tesforseq: -1, 0, 0, 2, -5, 10, -16, 24, -33, 44, -56, 70, -85, 102,
-120, 140,  

 1lesforseq: 1, 0, 0, 0, 1, -2, 4, -6, 9, -12, 16, -20, 25, -30, 36,
-42, 49, -56,  

 2jesforseq: 1, 0, 0, 0, 1, -2, 4, -6, 9, -12, 16, -20, 25, -30, 36,
-42, 49, -56,  

And we have our first "nontrivial" results:

 lesforseq = http://www.research.att.com/projects/OEIS?Anum=A002620
 Quarter-squares  

 tesforseq = http://www.research.att.com/projects/OEIS?Anum=A001859
Number of series-reduced planted trees with n+7 nodes and 3 internal
 nodes.

with vesfor = tesfor + lesfor + jesfor

In my mind, the sequences tes, les, and jes are three strains of braided
hair- and in this case we have chosen to strictly follow the strain
"lesfor" from the scalp down the first 3 turns, i.e. iterations, of the
braid .

Fourth iteration (les):

1vesforseq: 1, 0, 0, 0, 1, -2, 4, -6, 9, -12, 16, -20, 25, -30, 36, -42,
49, -56, 

 2tesforseq: -1, 0, 0, 0, 2, -7, 17, -33, 57, -90, 134, -190, 260, -345,
447, -567, 7 

 1lesforseq: 1, 0, 0, 0, 0, 1, -3, 7, -13, 22, -34, 50, -70, 95, -125,
161, -203, 252,  

 2jesforseq: 1, 0, 0, 0, 0, 1, -3, 7, -13, 22, -34, 50, -70, 95, -125,
161, -203, 252,  

2tesforseq = http://www.research.att.com/projects/OEIS?Anum=A045947
Triangles in open triangular matchstick arrangement (triangle minus
one side) of side n.

1lesforseq = http://www.research.att.com/projects/OEIS?Anum=A002623

Fifth iteration (les): 

1vesforseq: 1, 0, 0, 0, 0, 1, -3, 7, -13, 22, -34, 50, -70, 95, -125,
161, - 

 2tesforseq: -1, 0, 0, 0, 0, 2, -9, 26, -59, 116, -206, 340, -530, 790, 

 1lesforseq: 1, 0, 0, 0, 0, 0, 1, -4, 11, -24, 46, -80, 130, -200, 295, 

 2jesforseq: 1, 0, 0, 0, 0, 0, 1, -4, 11, -24, 46, -80, 130, -200, 295, 

2tesforseq = http://www.research.att.com/projects/OEIS?Anum=A082289
1lesforseq = http://www.research.att.com/projects/OEIS?Anum=A001752
 (the sixth iteration is also listed) 

I imagine that it would be important see what is happening to the
various g.f.'s in each case. However, instead of going into that now-
why don't we follow another strain of hair down from the scalp!

Step 1: Transform tesseq by forcing vesseq to take on its values. 
Step 2:   Repeat step one (replacing the words  "tesseq" with
"tesforseq" and "vesseq" with "vesforseq").

First iteration (tes) 

1vesforseq: 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 

 2tesforseq: -1, 2, -1, 2, -1, 2, -1, 2, -1, 2, -1, 2, -1, 2, -1, 2, -1,

 1lesforseq: 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0,

 2jesforseq: 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0,

Second iteration (tes)

1vesforseq: -1, 2, -1, 2, -1, 2, -1, 2, -1, 2, -1, 2, -1, 2, -1, 2, -1,
2, -1,  

 2tesforseq: -5, 10, -14, 19, -23, 28, -32, 37, -41, 46, -50, 55, -59,
64, -68, 73,  

 1lesforseq: 1, -2, 4, -5, 7, -8, 10, -11, 13, -14, 16, -17, 19, -20,
22,  

 2jesforseq: 1, -2, 4, -5, 7, -8, 10, -11, 13, -14, 16, -17, 19, -20,
22, -23, 25, 

Third iteration (tes)

1vesforseq: -5, 10, -14, 19, -23, 28, -32, 37, -41, 46, -50, 55, -59,
64, -68, 73,  

 2tesforseq: -13, 38, -76, 128, -193, 272, -364, 470, -589, 722, -868,
1028, -1201,  

 1lesforseq: 1, -6, 16, -30, 49, -72, 100, -132, 169, -210, 256, -306,
361, -420,  

 2jesforseq: 1, -6, 16, -30, 49, -72, 100, -132, 169, -210, 256, -306,
361, -420, 484,   

I will submit lesforseq later today.  
The bisection of lesfor is the sequence 1, 16, 49, 100,  a(n) = (3n+1)^2
( Superseeker gives 4*x^2+4*x+1+(x^4-2*x^3+2*x-1)*F(x) = 0 )

This message concludes with what was originally a mistake. I made it
after the 2nd les iteration above. Instead of transforming the sequence
1, 0, 0, 1, -1, 2, -2, 3, -3, 4, -4, 5, -5, 6, -6, 7, -7, 8, -8,   I
slipped up and tranformed the sequence 1, -1, 2, -2, 3, -3, 4, -4, 5,
-5, 6, -6, 7, -7, 8, -8.

"Third" iteration (les): 

1vesforseq: 1, -1, 2, -2, 3, -3, 4, -4, 5, -5, 6, -6, 7, -7, 8, -8, 9,
-9, 10, -10,  

 2tesforseq: -1, -2, 7, -13, 21, -30, 41, -53, 67, -82, 99, -117, 137,
-158, 181,  

 1lesforseq: 1, 0, -1, 3, -5, 8, -11, 15, -19, 24, -29, 35, -41, 48,
-55, 63, -71,  

 2jesforseq: 1, 0, -1, 3, -5, 8, -11, 15, -19, 24, -29, 35, -41, 48,
-55, 63, -71, 

1lesforseq = http://www.research.att.com/projects/OEIS?Anum=A024206
Expansion of x^2*(1+x-x^2)/((1-x^2)*(1-x)^2)   
2tesforseq is unlisted 

Sincerely, 
Creighton 