Weighted Three Directions OTA and Weighted Hexapolic Picture Automata

Two-dimensional hexagonal arrays seen on a triangular grid can be treated as two-dimensional representations of three-dimensional rectangular parallelepipeds. We are introducing weighted 3 directions on-line tessellation automata (W3OTA) and investigating formal power series on hexagonal pictures. These are functions that map hexagonal pictures to elements of a semiring and provide an extension of two-dimensional hexagonal picture languages to a quantitative setting.