[seqfan] Tiling growth sequences, useful for proofs?

[brad klee]

Hiya Seqfans,

Congrats to Sascha Kurz on A103465, it looks like you've beaten
us to the counts again. We've still got some nice illustrations here:

https://writings.stephenwolfram.com/2023/11/aggregation-and-tiling-as-multicomputational-processes/

However, it looks like at OEIS we're missing a similar sequence
for all the possible animals formed by famous pattern blocks:

[https://community.wolfram.com/groups/-/m/t/3061504](https://community.wolfram.com/groups/-/m/t/3061504?p_p_auth=CwWqZmq3)

The first few terms were: 6, 46, 1397. If these are of interest to any
one else--other than potentially kids everywhere who play with the
blocks in school--these numbers should be double checked and
extended before entry.

Perhaps sequences like this could be useful in studying proofs of
aperiodic tilings such as Penrose, Hat, Socolar 12-fold, T+C, etc.

But what we've found out recently, is that concentric ring-growth
search trees are better data structures, because they don't have
any branch mergers, and they don't allow worm-like configurations.
Both of these features help with the necessary outlaw proofs.

That said... It's very difficult to agree on canonical growth count
sequences, because boot-strapping the growth process involves
choices about which atlas rules to load and when.

I'm also writing (with two other enthusiasts) a follow up to Stephen
Wolfram's blog, which should be a proof of the Hat tiling's natural
realization in terms of combinatorial hexagons:

https://resources.wolframcloud.com/FunctionRepository/resources/HatHexagons/

It would be nice to get more sequences out of the calculations we've
done to prove the rules, but it's a totally new idea that will require more
thought. Nothing to follow in OEIS, as far as I know.

Ideas or suggestions from others are welcome (and I thought I saw
Joseph Myers on here the other day). The basic question is like this:

If we end up with two significantly different proofs for the hat tiling, then
how can we summarize the data models to say which proof is quantitatively
the less complex? Is it enough to collect a big set of search graphs and
then total their vertex counts?

All the best,

--Brad