Enumeration of directed site animals on two-dimensional lattices

Studies the problem of directed site animals on the square, triangular and hexagonal lattices. Closed form expressions are proposed for A(s), the number of animals of size s, on the square and triangular lattices. These expressions have been checked for s<or=33 and s<or=10 for the square and triangular lattices respectively by explicit enumeration. They imply that A(s) varies as lambda 2 s -0 for large s, where lambda =3 for the square lattice, and lambda =4 for the triangular, and theta = 1 / 2 for both. For the hexagonal lattice, A(s) is found for s