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
![kappa=lim_(n->infty)[F(n,n)]^(1/n^2)=1.503048082...](https://mathworld.wolfram.com/images/equations/HardSquareEntropyConstant/NumberedEquation1.svg){kind=small}
(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.
See also
(0,1)-Matrix, Hard Hexagon Entropy Constant
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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."
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Cite this as:
Weisstein, Eric W. "Hard Square Entropy Constant." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/HardSquareEntropyConstant.html