Hard Hexagon Entropy Constant 

Consider an  (0, 1)-matrix such as

(1)

for . Call two elements  adjacent if they lie in positions  and ,  and , or  and  for some . Call  the number of such arrays with no pairs of adjacent 1s. Equivalently,  is the number of configurations of nonattacking kings on an  chessboard with regular hexagonal cells.

The first few values of  for , 2, ... are 2, 6, 43, 557, 14432, ... (OEIS A066863).

The hard square hexagon constant is then given by

			(2)
			(3)

(OEIS A085851).

Amazingly,  is algebraic and is given by

(4)

where

			(5)
			(6)
			(7)
			(8)
			(9)
			(10)
			(11)

(Baxter 1980, Joyce 1988ab).

The variable  can be expressed in terms of the tribonacci constant

(12)

where  is a polynomial root, as

			(13)
			(14)
			(15)

(T. Piezas III, pers. comm., Feb. 11, 2006).

Explicitly,  is the unique positive root

(16)

where  denotes the th root of the polynomial  in the ordering of the Wolfram Language.

REFERENCES:

Baxter, R. J. "Hard Hexagons: Exact Solution." J. Physics A 13, 1023-1030, 1980.

Baxter, R. J. Exactly Solved Models in Statistical Mechanics. New York: Academic Press, 1982.

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

Joyce, G. S. "On the Hard Hexagon Model and the Theory of Modular Functions." Phil. Trans. Royal Soc. London A 325, 643-702, 1988a.

Joyce, G. S. "Exact Results for the Activity and Isothermal Compressibility of the Hard-Hexagon Model." J. Phys. A: Math. Gen. 21, L983-L988, 1988b.

Katzenelson, J. and Kurshan, R. P. "S/R: A Language for Specifying Protocols and Other Coordinating Processes." In Proc. IEEE Conf. Comput. Comm., pp. 286-292, 1986.

Sloane, N. J. A. Sequences A066863 and A085851 in "The On-Line Encyclopedia of Integer Sequences."