[seqfan] Re: No loop in view?

[Brad Klee]

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]
> >
> > --
> > Seqfan Mailing list - http://list.seqfan.eu/
> >
>
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