[seqfan] Re: Pentagonal tiling coordination sequence?

Richard J. Mathar mathar at mpia-hd.mpg.de
Mon Nov 21 18:42:55 CET 2022

aw> Date: Wed, 16 Nov 2022 21:23:49 -0500
aw> From: Allan Wechsler <acwacw at gmail.com>
aw> To: Sequence Fanatics Discussion list <seqfan at list.seqfan.eu>
aw> Subject: [seqfan] Pentagonal tiling coordination sequence?
aw> Start with a single regular pentagon in the Euclidean plane. The first
aw> entry in my proposed sequence is 1, because we start with 1 pentagon.
aw> On the other side of each of the edges of this seed pentagon, draw a new
aw> pentagon. This creates 5 new pentagons, so the second entry is 5.
aw> On the other side of each of the *unoccupied *edges of the new pentagons,
aw> draw a new pentagon. Some of them will intersect with each other, but for
aw> this sequence we consider that to be okay. This creates 20 distinct new
aw> pentagons, so the third entry is 20.
aw> On the other side of each of the unoccupied edges of these second-tier
aw> pentagons, draw a third-tier pentagon. You would think there would be 80 of
aw> these, but in fact they include ten coincident pairs, five very easy to
aw> see, and five a bit more obscure, so there are only 70 new pentagons at
aw> level 3.
aw> I did this all in my head, and my visualization skills are not up to doing
aw> level 4. There are fifteen sequences in OEIS starting with 1, 5, 20, 70,

As far as I can tell the total number of pentagons after 1,2,3,.. steps
is 1, 6, 26, 106,... 
illustrated in https://www.mpia-hd.mpg.de/~mathar/progs/pentag.pdf
which shows incrementally after each step the new sides added and
the full tiling with all pentagons present so far.
The rule is that there is a test for any of the existing pentagon sides
whether a clock-wise or counter-clock-wise pentagon is missing, which
is added if missing.
A pentagon is missing if at least one of its 5 sides is missing.

That means as far as counted each new pentagon adds 4 new sides
which spawns 4 pentagons in the next generation, leading to A020989 (conjectured).

The total number of pentagon sides grows as 5, 25, 105, 275,..
if I'm correct.

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