[seqfan] Re: No loop in view?

[Éric Angelini]

Hello Peter,
Thank you for the clever NOUGHT suggestion!
Best,
É.


> Le 23 févr. 2019 à 18:31, Brad Klee <bradklee at gmail.com> a écrit :
> 
> Yes, A039982 follows from either substitution:
> 
> Rule: {0->11, 1->01}, Axiom: 11
> Rule: {0->11, 1->10}, Axiom: 1
> 
> Then the proof of aperiodicity is easy.
> 
> The asymptotic ratio #(1):#(2)=2:1 requires a period
> of the form 3*p. If 3*p is even, then deflate to an odd
> period T=3*p/2^n. The inflation rules require a pattern
> of the form
> 
> . . . _1_1_1_1_1 . . .
> 
> which can only have odd period 1, a contradiction. Thus
> A039982 is aperiodic, as is the tag system sequence.
> 
> Aside from the Fibonacci word, these two systems
> remind me of the paper folding sequence, because
> the ratio of 1s to 0s turns out to be whole integer.
> 
> --Brad
> 
> 
>> On Fri, Feb 22, 2019 at 9:14 PM Allan Wechsler <acwacw at gmail.com> wrote:
>> 
>> I suspect that when you remove the initial 1,2,3,0, you get A039982. You
>> can get many such sequences as the limit of various production rules, and
>> the output is almost never periodic. Here, the production rules are
>> something like 0->11, 1->01, with some adjustment at the start of the
>> sequence.
>> 
>> 
>>> On Fri, Feb 22, 2019 at 2:58 PM Éric Angelini <bk263401 at skynet.be> wrote:
>>> 
>>> Hello SeqFans,
>>> while I was searching my old messy papers to answer a request
>>> from Alex Bellos, I've bumped into this idea yesterday:
>>> 
>>> "The chunks sizes of consonants squeezed between successive
>>> pairs of vowels are given by the sequence itself":
>>> 
>>> ONE, TWO, THREE, ZERO, ONE, ONE, ZERO, ONE, ZERO, ONE, ONE, ONE, ZERO,
>>> ONE, ONE, ONE, ZERO, ONE, ZERO, ONE, ZERO, ONE, ONE, ONE, ZERO, ONE, ZERO,
>>> ONE, ZERO, ONE, ONE, ONE, ZERO, ONE, ONE, ONE, ZERO, ONE, ONE, ONE, ZERO,
>>> ONE, ZERO, ONE, ZERO, ONE, ONE, ONE, ZERO, ONE, ONE, ONE, ZERO, ONE, ONE,
>>> ONE, ZERO, ONE, ZERO, ONE, ZERO, ONE, ONE, ONE, ZERO, ONE, ZERO, ONE, ZERO,
>>> ONE, ONE, ONE, ZERO, ONE, ZERO, ONE, ZERO, ONE, ONE, ONE, ZERO, ONE, ONE,
>>> ONE, ZERO, ONE, ONE, ONE, ZERO, ONE, ZERO, ONE, ...
>>> 
>>> The sequence obviously extends itself forever -- but does it
>>> enter in a loop at some point?
>>> 
>>> Jean-Marc Falcoz has computed 30,000 terms and found no loop
>>> (a third of the terms are 0s, two thirds are 1s). Does someone
>>> have an idea about the sequence entering at some point in a
>>> loop -- or never?
>>> Best,
>>> É.
>>> [a copy of this mail was sent to Jean-Marc Falcoz and Neil Sloane]
>>> 
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>>> Seqfan Mailing list - http://list.seqfan.eu/
>>> 
>> 
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