The hard-hexagons entropy constant is algebraic (see below z number).
The value is :
1.3954859724793027352295006635668880689541037281446611908174721561357608803586
977746898378730852754279026689685607685657184842212457119511639349818266947083
252547173794947534862281229126187281554340126162747356973585709823756812898414
948800016934903723995652094568253572538633572005211925074739811015138086289661
268136787831885630404682747107477204686894756657580905530270066675404962427719
060854536142216836296933016900330937276956621269398726823104923047442882514781
702966107270054292812280795061336321550953581179745072336957434963259935073449
490894249329307540816210555328068610619705545037955077580725537613858033619505
210958967729699416630942601615566925218549336476968551824281894615092855649748
501359906929152571833851080212811049755339847366927914398892041851355831303575
673710465224807454744982583885183287167357146092090743402851746571565499082292
999884612996137479952358336507860770516087879631202738350102895965881076822440
14681214726789035888008851819053742866660552775722734105313225337
Taken from
The Favorite mathematical constants of Steven Finch, Mathsoft Inc.
The constant is given by this (see z below)...
124 1/3
a := - --- 11
363
2501 1/2
b := ----- 33
11979
/ 31 1/3 // 2501 1/2 \1/3 / 2501 1/2 \1/3\\1/3
c := |1/4 - --- 11 ||----- 33 + 1| - |----- 33 - 1| ||
\ 242 \\11979 / \11979 / //
1/4 7/12
3 11
z1 := 3/44 ------------------------------------------------------------------
/ 31 1/3 // 2501 1/2 \1/3 / 2501 1/2 \1/3\\2/3
|1/4 - --- 11 ||----- 33 + 1| - |----- 33 - 1| ||
\ 242 \\11979 / \11979 / //
1/3 1/2 1/3 1/3 2/3 1/2 1/2 2
z2 := (1 - (1 - %1 ) + (2 + %1 + 2 (1 + %1 + %1 ) ) )
31 1/3 // 2501 1/2 \1/3 / 2501 1/2 \1/3\
%1 := 1/4 - --- 11 ||----- 33 + 1| - |----- 33 - 1| |
242 \\11979 / \11979 / /
1/3 1/2 1/3 1/3 2/3 1/2 1/2 2
z3 := (- 1 - (1 - %1 ) + (2 + %1 + 2 (1 + %1 + %1 ) ) )
31 1/3 // 2501 1/2 \1/3 / 2501 1/2 \1/3\
%1 := 1/4 - --- 11 ||----- 33 + 1| - |----- 33 - 1| |
242 \\11979 / \11979 / /
1/3 1/2
z4 := 1/(1/33 (1089 + 372 11 )
/ 124 1/3 / 124 1/3 15376 2/3\1/2\1/2
+ |2 - --- 11 + 2 |1 - --- 11 + ------ 11 | | )^1/2
\ 363 \ 363 131769 / /
1/4 7/12
z := 3/44 3 11
1/3 1/2 1/3 1/3 2/3 1/2 1/2 2
(1 - (1 - %1 ) + (2 + %1 + 2 (1 + %1 + %1 ) ) )
1/3 1/2 1/3 1/3 2/3 1/2 1/2 2 /
(- 1 - (1 - %1 ) + (2 + %1 + 2 (1 + %1 + %1 ) ) ) / (
/
2/3 1/3 1/2
%1 (1/33 (1089 + 372 11 )
/ 124 1/3 / 124 1/3 15376 2/3\1/2\1/2
+ |2 - --- 11 + 2 |1 - --- 11 + ------ 11 | | )^1/2)
\ 363 \ 363 131769 / /
31 1/3 // 2501 1/2 \1/3 / 2501 1/2 \1/3\
%1 := 1/4 - --- 11 ||----- 33 + 1| - |----- 33 - 1| |
242 \\11979 / \11979 / /
> evalf(z);
1.395485972479302735229500663566888068954103728144661190817472165