[seqfan] More dragon conjectures

[brad klee]

Hiya seqfans,

Nice work on A005811, but did anyone notice that it's also the
recursion depth of A035327?

https://oeis.org/draft/A005811

https://oeis.org/A035327

This suggests to obtain Dragon analogs by base transformation,
and here's a sample of what I found:

https://oeis.org/A043555
1, 1, 1, 2, 1, 2, 2, 2, 1, 2, 3, 3, 2, 1, 2, 3, 3, 2, 2, 3, 3 . . .
(possibly better if the first term is zero)

try2b[n_] := Map[Function[x, Length[Most[NestWhileList[
FromDigits[Mod[IntegerDigits[#, 3] - 1, 3], 3] &, x, UnsameQ[#, 0] &]]]], Range[0, n]]

0, 1, 2, 3, 1, 2, 3, 4, 2, 3, 4, 5, 3, 1, 2, 3, 4, 2, 3, 4, 5 . . .

First differences for both of these sequences appear to be
limit periodic with block length 9, and also with the last column
being an exact copy of the entire sequence. Differences of
A043555 are written over {1,0,-1}, while try2b is over {1, -2}.
There is likely a projection map between the two.

Joerg pointed out Gheorghe's graceful form, but changing the
base of it did not get any cool graphics off ground.

Under the usual graphical map onto Eisenstein integers, A043555
appears, not to intersect on edges and to cover arbitrarily large disks.
The try2b is just a triangle, but that is also noteworthy (from a "No
dragon Zen" point of view). It amounts to an extra very simple
constraint on values.

https://0x0.st/XaEy.png

If the conjectures are true, then we have a dragon that
transforms into a triangle, and can also investigate the
inverse transform, like yin and yang.

Does anyone have a pre-existing reference for the graphical
observation? It could have been known since 1999...

All the best,

--Brad