on naming a sequence

[Creighton Dement]

> Date: Fri,  3 Jun 2005 16:56:17 +0200
> Subject: Re: on naming a sequence
> From: Jon Awbrey <jawbrey at att.net>
> To: seqfan at ext.jussieu.fr

> o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
> 
> Creighton Dement wrote:
> > 
> > Dear Seqfans,
> > 
> > Only after I began the submission process for
> > the two top sequences listed (and graphed) at:
> > http://www.crowdog.de/Triton/SnakeCharmer/SnakeCharmer.html
> > did I realize I have no idea on how to label the name field.
> > The program> FAMP has no trouble finding a name for them --
> > namely 1em[I]cyczapsumrokseq  and 1em[I*]cyczapsumrokseq,
> > respectively.  However, I don't want to write something as
> > unseemly as that into the name field.  (I also contemplated
> > mentioning in detail what  each of the syllables em[I], cyc,
> > zap, sum, rok in its name does- that would take up a lot of
> > space unless it was kept very brief).
> >
> <...>
> 
> it might help if those and other defs
> were placed at some core or root seq
> in the oeis, or a linked txt file?
> 
> ja
> 
> o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
> inquiry e-lab: http://stderr.org/pipermail/inquiry/
> o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o


That might be a good idea. I find both sequences baffling. Concerning
the first sequence: Using a program to search through the first 3000
terms, I uncovered a few additional rules last night which, in some
cases, allow the next term of the sequence to be calculated (to be
precise, I should say observations were made which appear to be rules) .



The rules are given, below (if I write ?= it means that not enough terms
were found to really know with confidence).  
 
Rule I: the range of the sequence is {0, +-1, +-2, +-3}

Rule II: apart from the initial term, the sequence follows the pattern
odd, odd, odd, even, even, even or (or a(m) is even if and only if m is
congruent to {0, 4, 5} mod 6)

Rule III: 
Let (a_n) be the sequence.  
Let cxc' stand for the set of all numbers y such that numbers c, c' and
m  exist with a(m) = c, a(m+1) = y, and a(m+2) = c'. 

(a) -2x2 = 2x-2 = empty set 

(b) 2x0 = -2x0 = 0x2= 0x-2 = {0}

(c) 0x0 = 2x2 = -2x-2 = {+2, -2}
 
(d) 1x0  = -3x0 = 0x-3 = -1x-2 = -2x-1 =   2x3 = 3x2 = {0, 1, +-2, -3}

(e) -1x0 = 0x3  = 3x0  = 1x2  = 2x1  = -2x-3  = -3x-2 = {0, -1, +-2, 3}

(f) 2x-3 = -3x2 = -2x3 = 3x-2 = {+-2, 0}

(g) -2x1 ?= {0, -1, +-2}

(h) 1x-2 ?=  {0, +-2, 3} 
 
(i) 1x1 = 1x-1 = -1x1 = -1x-1 = -3x1 = 1x-3 = 3x3 = 3x-3 = -3x3 = -3x-3
= -1x-3 = -3x-1 = {+-1, +-3} 

(j) -1x3 ?= {1, 3}   

(k) -1x2 ?= {2,0}  

(l) = 2x-1 = {+-2,0} 


It seems natural to ask (for ex., given a(n), a(n+1), a(n+2)):  under
what conditions do the rules uniquely determine a(n+3).  

Starting with 0, 1, -3, 3, -2, 2, -2, -3, y, z, p ;  assume we are given
the above rules and want to know y and z. 
from (a), y != -2, y != 2  ; follows equivivalently from Rule II. 
from (b), y != 0 ; follows equivivalently from Rule II. 
from (f),  y != 1
from (d), y could be -1
from (f), y != 3
from (e), y != -3

Thus, y = -1. 

0, 1, -3, 3, -2, 2, -2, -3, -1, z  

from (d), z != 0 ; follows equivivalently from Rule II
from (f), z != 2 ; follows equivivalently from Rule II
from (e), z could be -2 but from Rule II, z != -2
from (i), z could be 1 or -1 or 3 or -3


Where do we go from here?
Assume z = 3, then 
0, 1, -3, 3, -2, 2, -2, -3, -1, 3, p

By Rule II, p = +-2 or p = 0. 
Assume p = 2,  contradicts (k) 
Assume p = -2, contradicts (d)
Assume p = 0, no contradiction


Assume z = 1, then 
0, 1, -3, 3, -2, 2, -2, -3, -1, 1, p

By Rule II, p = +-2 or p = 0.
Assume p = 0, contradicts (e)
Assume p = 2, contradicts (k)
Assume p = -2  no contraction

It follows that there no rules seem to directly forbid z = 1, or z = 3.
But the sequence gives z = -3... could there be a missing rule?


Sincerely, 
Creighton