Conway and Lagarias showed that certain roughly triangular regions in the hexagonal grid cannot be tiled by shapes Thurston later dubbed tribones. Here we introduce a two-parameter family of roughly hexagonal regions in the hexagonal grid and show that a tiling by tribones exists if and only if the two parameters associated with the region are the paired pentagonal numbers $k(3k \pm 1)/2$.
@article{10_37236_11326,
author = {Jesse Kim and James Propp},
title = {A pentagonal number theorem for tribone tilings},
journal = {The electronic journal of combinatorics},
year = {2023},
volume = {30},
number = {3},
doi = {10.37236/11326},
zbl = {1533.05052},
url = {http://geodesic.mathdoc.fr/articles/10.37236/11326/}
}
TY - JOUR
AU - Jesse Kim
AU - James Propp
TI - A pentagonal number theorem for tribone tilings
JO - The electronic journal of combinatorics
PY - 2023
VL - 30
IS - 3
UR - http://geodesic.mathdoc.fr/articles/10.37236/11326/
DO - 10.37236/11326
ID - 10_37236_11326
ER -
%0 Journal Article
%A Jesse Kim
%A James Propp
%T A pentagonal number theorem for tribone tilings
%J The electronic journal of combinatorics
%D 2023
%V 30
%N 3
%U http://geodesic.mathdoc.fr/articles/10.37236/11326/
%R 10.37236/11326
%F 10_37236_11326
Jesse Kim; James Propp. A pentagonal number theorem for tribone tilings. The electronic journal of combinatorics, Tome 30 (2023) no. 3. doi: 10.37236/11326
