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.
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."
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).
![kappa=lim_(n->infty)[F(n,n)]^(1/n^2)=1.503048082...](/images/equations/HardSquareEntropyConstant/NumberedEquation1.svg)
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.
