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Reference Philippe Flajolet and Andrew Odlyzko in Random Mapping Statistics you can have the article at ftp://netlib.att.com/netlib/att/math/odlyzko/index.html 1 - ln(1 - 1/E)
> evalf(",1024);
1.4586751453870818910216436450673297018769779066921941448349981657928142090774\ 201612200442809516952542077265289812147224950456505217508488257192318776903978\ 283958471454981649855439295026537053597338520354935148025543820985296873219986\ 302608076828991375664708977028227357407155020168390466081440332929613402809962\ 987761600422067245386552208829277426092542078462258992350164685882837621214882\ 780180315165656808973787662538495808236640442271087689278355793100958663124347\ 608912549488795731777070799343730722066801620056545869945636645492898791927486\ 575158188313946857834776772734408679626984363705284330037652725380287794676349\ 373789251316549424606319247455867160631085208147788915528328222030175460874293\ 072958579419651653681072447431245769874928136318703222181432813223236987618651\ 560035148342838332185451812183617068075562954967559891795834498316055598164437\ 208325189384466039982301475617199617179127040273935951240040637361969048372804\ 683416371677229307327020903657448359390542480371335759362920019292630614667717\ 96831954446
1 + exp(-1) —————- exp(-1) - 1
-2.163953413738652848770004010218023117093738602150792272533574120
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The Hard hexagons Entropy Constant
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 fromThe Favorite mathematical constants of Steven Finch, Mathsoft Inc.
The constant is given by this (see z below)…
124 1/3a := - —- 11363
2501 1/2b := ——- 3311979
/ 31 1/3 // 2501 1/2 \1/3 / 2501 1/2 \1/3\\1/3c := |1/4 - —- 11 ||——- 33 + 1| - |——- 33 - 1| ||\ 242 \\11979 / \11979 / //
1/4 7/123 11z1 := 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 2z2 := (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 2z3 := (- 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 1/2z4 := 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 7/12z := 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
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End of Project Gutenberg's Miscellaneous Mathematical Constants, by Various