Complexity of cutting words on regular tilings

We show that the complexity of a cutting word uu in a regular tiling with a polyomino QQ is equal to Pn(u)=(p+q−1)n+1Pn(u)=(p+q−1)n+1 for all n≥0n≥0, where Pn(u)Pn(u) counts the number of distinct factors of length nn in the infinite word uu and where the boundary of QQ is constructed of 2p2p horizontal and 2q2q vertical unit segments.