Dean¹s Word: Omega 2,1

[Robert G. Wilson v]

Dear Jeremy,

	Outside of the OEIS, I probably have the largest set of morphisms around and
you are correct, the trajectory is not in the OEIS.

Nest[Flatten[# /. {0 -> {0, 1}, 1 -> {2, 1}, 2 -> {0, 3}, 3 -> {2, 3}}] &, {0}, 7]
{0, 1, 2, 1, 0, 3, 2, 1, 0, 1, 2, 3, 0, 3, 2, 1, 0, 1, 2, 1, 0, 3, 2, 3, 0, 1, 2, 3, 0, 3, 2, 1, 0, 1, 2, 1, 0, 3, 2, 1, 0, 1, 2, 3, 0, 3, 2, 3, 0, 1, 2, 1, 0, 3, 2, 3, 0, 1, 2, 3, 0, 3, 2, 1, 0, 1, 2, 1, 0, 3, 2, 1, 0, 1, 2, 3, 0, 3, 2, 1, 0, 1, 2, 1, 0, 3, 2, 3, 0, 1, 2, 3, 0, 3, 2, 3, 0, 1, 2, 1, 0, 3, 2, 1, 0, 1, 2, 3, 0, 3, 2, 3, 0, 1, 2, 1, 0, 3, 2, 3, 0, 1, 2, 3, 0, 3, 2, 1}


or the concatenation of the stages.
NestList[Flatten[# /. {0 -> {0, 1}, 1 -> {2, 1}, 2 -> {0, 3},  3 -> {2, 3}}] &, {0}, 6] // Flatten
{0, 0, 1, 0, 1, 2, 1, 0, 1, 2, 1, 0, 3, 2, 1, 0, 1, 2, 1, 0, 3, 2, 1, 0, 1, 2, 3, 0, 3, 2, 1, 0, 1, 2, 1, 0, 3, 2, 1, 0, 1, 2, 3, 0, 3, 2, 1, 0, 1, 2, 1, 0, 3, 2, 3, 0, 1, 2, 3, 0, 3, 2, 1, 0, 1, 2, 1, 0, 3, 2, 1, 0, 1, 2, 3, 0, 3, 2, 1, 0, 1, 2, 1, 0, 3, 2, 3, 0, 1, 2, 3, 0, 3, 2, 1, 0, 1, 2, 1, 0, 3, 2, 1, 0, 1, 2, 3, 0, 3, 2, 3, 0, 1, 2, 1, 0, 3, 2, 3, 0, 1, 2, 3, 0, 3, 2, 1}


Sincerely, Bob.

P.S. Neil, I am assuming that you wish this to be written up. Bob.


Jeremy Gardiner wrote:

> On reading the following lecture notes:
> 
> http://www.latrobe.edu.au/mathstats/maths/conferences/vic_algebra/CombonWords.pdf
> 
> Avoidable Words
> George F. McNulty
> University of South Carolina
> 2003
> 
> I found the following sequence:
> 
> Dean¹s Word: Omega 2,1
> 
> Let phi be the endomorphism given by
> 0 -> 01
> 1 -> 21
> 2 -> 03
> 3 -> 23
> 
> Start from 0 and iterate phi :
> 
> 0
> 01
> 0121
> 01210321
> 0121032101230321
> 
> Ref. appears to be:
> 
> Dean, Richard A. 1965. A sequence without repeats on x, x € 1 , y, y € 1 ,
> Amer. Math. Monthly 72 , 383­ 385. MR 31 #3500
> 
> - The sequence doesn't appear to be in the OEIS but I wonder if it 
> appears in another guise?
> If nobody recognises this as already in the OEIS, I will submit the 
> sequence.
> 
> Jeremy Gardiner