# words-length and words in a group

the group is $\mathbb{Z}_2*\mathbb{Z}_3=\langle a,\ b\mid a^2,\ b^3\rangle$, and the picture:

Correctly predicts the next one. It is A164001 in the OEIS data-base. Dubbed “Spiral of triangles around a hexagon“. It has the generating function $-(x+1)+\frac{x^2+2x+1}{1-x^2-x^3}$, Why would it be? :) . This another is A000931.