# [seqfan] Re: Pentagonal tiling coordination sequence?

Neil Sloane njasloane at gmail.com
Sun Nov 20 15:49:26 CET 2022

```I have worked on coordination sequences since at least the 1990s (see
A008000 for example), and I have never heard of "layer sequence".  It only
appears in 5 seqs, I think, and all were submitted by the same person.

We should certainy add a comment to them saying that they are a kind of
C.S., and also mention them in the OEIS Index under the entry for Coord
Seqs!

Best regards
Neil

Neil J. A. Sloane, Chairman, OEIS Foundation.
Also Visiting Scientist, Math. Dept., Rutgers University,
Email: njasloane at gmail.com

On Sun, Nov 20, 2022 at 8:42 AM Allan Wechsler <acwacw at gmail.com> wrote:

> Start with a single regular pentagon in the Euclidean plane. The first
> entry in my proposed sequence is 1, because we start with 1 pentagon.
>
> On the other side of each of the edges of this seed pentagon, draw a new
> pentagon. This creates 5 new pentagons, so the second entry is 5.
>
> On the other side of each of the *unoccupied *edges of the new pentagons,
> draw a new pentagon. Some of them will intersect with each other, but for
> this sequence we consider that to be okay. This creates 20 distinct new
> pentagons, so the third entry is 20.
>
> On the other side of each of the unoccupied edges of these second-tier
> pentagons, draw a third-tier pentagon. You would think there would be 80 of
> these, but in fact they include ten coincident pairs, five very easy to
> see, and five a bit more obscure, so there are only 70 new pentagons at
> level 3.
>
> I did this all in my head, and my visualization skills are not up to doing
> level 4. There are fifteen sequences in OEIS starting with 1, 5, 20, 70,
> and none of them are obviously this pseudo-tiling coordination sequence,
> although some of them might turn out to be. One of them is the coordination
> sequence of a closely related tiling, a hyperbolic tiling with Schläfli
> symbol {5,5}. It's barely plausible that there is an isomorphism of some
> sort between these two tilings, in which case https://oeis.org/A054889 is
> indeed the sequence I'm looking for, and we should just add a comment that
> explains this.
>
> Oh, apparently coordination sequences of this kind are called layer
> sequences. I find it odd that we have two different terms for concepts so
> closely related. For any layer sequence, there is clearly a dual graph that
> has the same numbers as its coordination sequence. But hey, there's lots of
> odd mathematical nomenclature.
>
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>

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