## Hard Hexagon Entropy Constant

N.B. A detailed on-line essay by S. Finch was the starting point for this entry.

A constant related to the Hard Square Entropy Constant. This constant is given by

 (1)

where is the number of configurations of nonattacking Kings on an chessboard with regular hexagonal cells, where . Amazingly, is algebraic and given by
 (2)

where
 (3) (4) (5) (6) (7) (8) (9)

(Baxter 1980, Joyce 1988).

References

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

Finch, S. Favorite Mathematical Constants.'' http://www.mathsoft.com/asolve/constant/square/square.html

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

Plouffe, S. Hard Hexagons Constant.'' http://www.lacim.uqam.ca/piDATA/hardhex.html.