Hard Square Entropy Constant

Author

Eric W. Weisstein
April 16, 2004

This notebook downloaded from http://mathworld.wolfram.com/notebooks/Constants/HardSquareEntropyConstant.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/HardSquareEntropyConstant.html.

©2005 Wolfram Research, Inc. except for portions noted otherwise

Sequence

fs={2,7,63,1234,55447,5598861,1280128950,660647962955,770548397261707,2030049051145980050,12083401651433651945979, 162481813349792588536582997,4935961285224791538367780371090,338752110195939290445247645371206783};

κs=MapIndexed[#1^(1/#2[[1]]^2)&,fs]

κs//N

Constant

κ=SetPrecision[1.503048082,10]

SequenceLimit[N[κs,50]]

Plot

<<Utilities`Typesetting`

ListPlot[κ,
AxesLabel->TraditionalForm/@{n,F[n,n]^InlineFraction[1,n^2]},
Joined->True,PlotStyle->Red,
Epilog->{Red,PointSize[.02],MapIndexed[Point[{#2[[1]],#1}]&,κ]},
PlotRange->All,
AxesOrigin->{1,1.5},
PlotRange->{{-.9,14.1},{1.4,2}}]