[seqfan] Re: Pentagonal tiling coordination sequence?

Allan Wechsler acwacw at gmail.com
Tue Nov 22 00:06:54 CET 2022

I believe Richard Mathar's analysis to be incorrect, because he is
essentially counting paths, and some pairs of paths lead to the same
pentagon in the third layer. I fear, however, that the full discussion is
too detailed and finicky to interest the whole list. I invite people who
want to participate to contact me at acwacw at gmail.com.

On Mon, Nov 21, 2022 at 12:43 PM Richard J. Mathar <mathar at mpia-hd.mpg.de>

> 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>
> 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>
> 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>
> 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>
> 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>
> 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,
> aw>
> 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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