[seqfan] Automatons on the Chair Tiling

[Brad Klee]

Hi Seqfans,

Probably the fractal structure of A319018 can by proven by
an analogy with C.A. Rule 90. While exploring this direction,
I discovered A147562, and quickly computed the following:

https://oeis.org/A147562/a147562_1.png

which superimposes a log-periodic coloring of the ON cells
above the chair tiling. It shows pretty clearly that ON cells
are one-to-one with chair tiles, as has already been noted.

A319018 has the same linear inflation factor 2, and shows
imperfect symmetry-resonance with the chair tiling:

https://oeis.org/A319018/a319018_3.png

Lack of 1-to-1 correspondence gives some idea as to why
A322050 is so much more difficult than A147610.

Pay close attention to locations where four chairs meet at
one corner to form a cross. Assume one isolated ON site
surrounded by eight more isolated ONs--is this group
necessarily drawn above a chair cross to preserve local
D_8 symmetry?

Such symmetry points are also critical "branch points" in
the aforementioned proof, so an affirmative answer would
be something incredible.

Cheers,

--Brad