Let F(m, n) be the number of m×n (0, 1)-matrices with no adjacent 1s (in either columns or rows). For n = 1, 2, ..., F(n, n) is given by 2, 7, 63, 1234, ... (OEIS A006506). The hard square entropy constant is defined by κ congruent lim_(n->∞) [F(n, n)]^(1/n^2) = 1.50305... (OEIS A085850). It is not known if this constant has an exact representation. The quantity ln κ arises in statistical physics (Baxter et al. 1980, Pearce and Seaton 1988), and is known as the entropy per site of hard squares. A related constant known as the hard hexagon entropy constant can also be defined.

(0, 1)-matrix | hard hexagon entropy constant