A Pentagonal Number Theorem for Tribone Tilings

[摘要] 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 $k(3k \pm 1)/2$.

[学科分类] 统计和概率