Hard Square Entropy Constant

Let  be the number of  (0,1)-matrices with no adjacent 1s (in either columns or rows). For , 2, ...,  is given by 2, 7, 63, 1234, ... (OEIS A006506).

The hard square entropy constant is defined by

(OEIS A085850). It is not known if this constant has an exact representation.

The quantity  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.

REFERENCES:

Baxter, R. J.; Enting, I. G.; and Tsang, S. K. "Hard-Square Lattice Gas." J. Statist. Phys. 22, 465-489, 1980.

Finch, S. R. "Hard Square Entropy Constant." §5.12 in Mathematical Constants. Cambridge, England: Cambridge University Press, pp. 342-349, 2003.

Pearce, P. A. and Seaton, K. A. "A Classical Theory of Hard Squares." J. Statist. Phys. 53, 1061-1072, 1988.

Sloane, N. J. A. Sequences A006506/M1816 and A085850 in "The On-Line Encyclopedia of Integer Sequences."