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, ... (Sloane's A006506).

The hard square entropy constant is defined by

(Sloane's 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.

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