[seqfan] Re: The interest value of A019989, A019990, and A019991

[Brad Klee]

Hi Peter,

No, there are no oscillations. The counts increase exponentially
toward asymptotic relative
density, and it looks like disparity persists. With the newly proposed
definition, It's actually
possible to calculate the principle eigen-vector, but I haven't the
time just now.

On A287451 and on Kimberling's other ternary examples, equal density
is guaranteed by
the fact that each block of three letters is some permutation of 012.
This property then
ensures that in-column permutation property of the associated index
sequences (which
I mentioned earlier).

Gosper uses rules on six symbols, but with only three letters per
replacement rule. If the
design goal is to reach equal density, then the rules could look something like:

a->BaCcAb, A->cAbBac, etc...

To explore every such well balanced replacement, as Kimberling was
doing with n=3, would
lead us into a function space on the order of (6!)^6 . . . This is too
many, more-or-less random
sequences of digits, especially considering that we have yet to figure
out if this Gosper
sequence has any deeper mathematical meaning or is it meant to be more
a work of Art
in the genre of alphabet soup?

Cheers,

Brad



On Thu, Apr 18, 2019 at 1:13 PM Peter Munn <techsubs at pearceneptune.co.uk> wrote:
>
> Dear seqfans,
>
> On Tue, April 16, 2019 3:12 am, Brad Klee wrote:
> > On Mon, Apr 15, 2019 at 10:30 AM Marc LeBrun <mlb at well.com> wrote:
> >> I was hoping it was trying to be some kind of ternary Thue-Morse
> > Yes, it does look something similar to a ternary Thue-Morse, but
>
> [...]
> > Notice that this analogy preserves that each column should
> > contain a permutation of three consecutive numbers. This
> > property does not hold for A019989, A019990, A019991.
>
> I suspect we should look at the densities of each sequence plotted against
> log N, where the density is measured over 1..N . I would expect to see
> oscillations with peaks and troughs that may not converge, but the
> respective asymptotes for the peaks and the troughs might be the same for
> each of the three sequences.
>
> Best Regards,
>
> Peter  (with apologies for any nonstandard terminology from this amateur)
>
>
>
>
>
>
>
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