A pentagonal number theorem for tribone tilings

Conway and Lagarias showed that certain roughly triangular regions in the hexagonal grid cannot be tiled by shapes Thurston later dubbed tribones. Here we study 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 $k(3k \pm 1)/2$.