# [seqfan] Coord. seqs., the coloring book approach; 2-uniform tilings

Neil Sloane njasloane at gmail.com
Tue Mar 27 20:54:08 CEST 2018

```Dear SeqFans, Chaim Goodman-Strauss and I have written up our coloring book
method for finding coordination seqs, see
"A Coloring Book Approach to Finding Coordination Sequences",
https://arxiv.org/abs/1803.08530

The motivation was to find simple proofs for various conjectured formulas
in the OEIS (see for example A250120).

The latter is one of the 11 uniform tilings, see

List of coordination sequences for uniform planar nets: A008458(the planar
net 3.3.3.3.3.3), A008486 (6^3), A008574 (4.4.4.4 and 3.4.6.4), A008576
(4.8.8), A008579 (3.6.3.6), A008706(3.3.3.4.4), A072154 (4.6.12), A219529
(3.3.4.3.4), A250120(3.3.3.3.6), A250122 (3.12.12).
These now all have g.f.s.

I've just added all 20 2-uniform tilings to the OEIS, see

Coordination sequences for the 20 2-uniform tilings in the order in which
they appear in the Galebach catalog, together with their names in the RCSR
database (two sequences per tiling): #1 krt A265035, A265036; #2 cph
A301287, A301289; #3 krm A301291, A301293; #4 krl A301298, A298024; #5 krq
A301299, A301301; #6 krs A301674, A301676; #7 krr A301670, A301672; #8 krk
A301291, A301293; #9 krn A301678, A301680; #10 krg A301682, A361684; #11
bew A008574, A296910; #12 krh A301686, A301688; #13 krf A301690, A301692;
#14 krd A301694, A219529; #15 krc A301708, A301710; #16 usm A301712,
A301714; #17 krj A219529, A301697; #18 krc A301716, A301718; #19 krb
A301720, A301722; #20 kra A301724, A301726.

Many of these have only the 10 terms that RCSR gives.  If anyone has access
to ToposPro, that will give 127 terms (the program is free but requires a
Windows machine).  If we had more terms we could then probably guess a g.f.
using gfun.

Even without more terms, our coloring book method should easily produce
recurrences in most cases, if anyone wants to try it. It is also great fun
to play with - we were originally going to call the paper "A Child's
Coloring Book Approach to Finding Coordination Sequences". But children are
not essential, all you need are colored pencils.
```