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Miscellaneous Mathematical Constants, Equations, Derivations...

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/ 19 1/2\1/3 4
|---- + 1/9 33 | + ----------------------- + 1/3
\ 27 / / 19 1/2\1/3
9 |---- + 1/9 33 |
\ 27 /

in fact the n'th Tribonacci number is given by this EXACT formula.
---------------------------------------------------------------------

See : http://www.labri.u-bordeaux.fr/~loeb/book/92pl.php

Comment calculer le nieme nombre de Tribonacci

Resume of a conference given in 1993 (Universite Bordeaux I, LaBRI).

1/2 1/3 1/2 1/3 n 1/2 1/3
(1/3 (19 + 3 33 ) + 1/3 (19 - 3 33 ) + 1/3) (586 + 102 33 )
3 ---------------------------------------------------------------------------
1/2 2/3 1/2 1/3
(586 + 102 33 ) + 4 - 2 (586 + 102 33 )

To get the actual n'th Tribonacci number just round the result to the
nearest integer.

Here is the formula 'lprinted'...

3*(1/3*(19+3*33^(1/2))^(1/3)+1/3*(19-3*33^(1/2))^(1/3)+1/3)^n/((586+102*33^(1
/2))^(2/3)+4-2*(586+102*33^(1/2))^(1/3))*(586+102*33^(1/2))^(1/3);

This formula has 2 parts, first the numerator is the root of (x^3-x^2-x-1)
no surprise here, but the denominator was obtained using LLL (Pari-Gp)
algorithm. The thing is, if you try to get a closed formula by doing
the Z-transform or anything classical, it won't work very well since
the actual symbolic expression will be huge and won't simplify.

The numerical values of Tribonacci numbers are c**n essentially and
the c here is one of the roots of (x^3-x^2-x-1), then there is another
constant c2. So the exact formula is c**n/c2.

Another way of doing 'exact formulas' are given by using [ ] function
the n'th term of the series expansion of 1/(1+x+x**2) is

1 - 2 floor(1/3 n + 2/3) + floor(1/3 n + 1/3) + floor(1/3 n)
-----------------------------------------------------------------------------
The twin primes constant.

0.660161815846869573927812110014555778432623
-----------------------------------------------------------------------------
The Varga constant, also known to be the 1/(one-ninth constant).

9.2890254919208189187554494359517450610317

One-ninth constant is 0.1076539192264845766153234450909471905879765038
-----------------------------------------------------------------------------
0.4749493799879206503325046363279829685595493732172029822833310248
6455792917488386027427564125050214441890378494262395464775250455
2099778523950882780814821592082565202912193041770281959987798787
6404342380353179170625016170252803841553681975679189489592083858

to 256 digits is also this closed expression.

2**(5/4)*sqrt(Pi)*exp(Pi/8)*GAMMA(1/4)**(-2);

-----------------------------------------------------------------------------
-Zeta(1,1/2).

is also equal to -Zeta(1/2)*(1/2*gamma+1/2*ln(8*Pi)+1/4*Pi).

3.922646139209151727471531446714599513730323971506505209568298485
2547208031503382848806505231041456914038034379886764996843321856
0187370796648866325531877003002927708284792679262934379740474743
4560678349258709176744625306684542186046544092107149397014020908

-----------------------------------------------------------------------------
-Zeta(-1/2) to 256 digits.
0.2078862249773545660173067253970493022262685312876725376101135571
0614729193229234048754326694073321564310997561412868956566132691
4694458311965705623294109531061640017807007041375078320755666248
7877869206615046914282912338325693716136777293836109459387888090

-----------------------------------------------------------------------------
Zeta(2) or Pi**2/6 to 10000 places.

1.6449340668482264364724151666460251892189499012067984377355582293700074704032
008738336289006197587053040043189623371906796287246870050077879351029463308662
768317333093677626050952510068721400547968115587948903608232777619198407564558
769632356367097100969489020859320080516364788783388460444451840598251452506833
876314227658793929588063204472197908477340910590208378289549278263890379763583
343942045159120818099593454448774587965008808894087011116347106931614618428879
815486244835909183448757387428394082760287563214346010013576620982048720690400
073826635603024022844629630324566097171951427721315951255679986190871931543953
524106380440721421339654750580158723165839947624349142243348362904887009665059
862263034109596736552811371670326911498784034357161605776676333067252736894238
416640889536227595400772794748127102520498378433230017165744810302860434966884
794216728433597281997793810008466560780537782885947278625931618664588292160658
193859232415325806461781201884649777625984977560609384606051467685834725623197
101836301479837488962159297027632358745738223006797795679319515651996612383618
366168655665797003758579395038193467059393114859491596635058620858526381064548
879582000789743717215693657490825080352045741139287635530947709860823922939866
707500525803645340315412739072742722890227479742157521265272866790504356086447
019522174348296308095407209404388845394174205278719269341962282024749751511874
134727875179936647336874820752335660885793907659619607908126511591050729219558
844613572641252614751578071609175156885327683293665654765588128436115113494859
670092266296975220677781810295008702914015225183747431377217755317906719967001
114954768292364207502705341165049051072861188854707754573575854747032957919907
087156125812402558853000196898875722439717953811180793070896494335953356183275
794651103546695668292833094507406208425346300827605686180238175238239659462458
207920249063737872085300479379967603565543851521312093605893490413075491311959
041935877531888380567912171377264570722995635142812810658216832092872867483537
830128254732917028021436897618019637363184980566899586355341068647425930801883
367749469866838428949777402705311753583758607474169405737637153525165870187112
803861643246178480126671392369158545043444646648471950875283006191625838679257
789892298444165212547711817391890576286084578861368469335293800824741929432439
323626468769086749231576094206150249840056930228249239061832435185795019030056
175145835716574335223282351666140476391283940576264724881002052041812033788626
252366555788937763981538291415976032314805706590691944583703140205153805821921
917295905553978794000789946119846527541470685853650059963662675570615695254317
725315543876351727626192497886160965070445249636970214088526846826793337277335
304510049048440997806284485082110994618287767772335914596367843974103130509587
129133090687474711615266669048005032852464489328907980999569273302366947444696
980408971357140919211944272389692435581378274354272335097373721967750814686576
663441952446411700179575195463240270033858392729754878763962111770937842133454
352824431047423526125821320401230363123983273324026549756391540540044136979906
766267255921507827975082470078437497364993986919795895302438696540685316220558
806647674709744582710116043808080019623575571358222189260692237675032277565352
064008617178017803580708485672571477572333511220120417258731709054252729576117
157856361679235388523430675984088509039181941283572915859606859411517352815919
870332098540687510589658192905746197996012164173315499267877030429691916001924
484991596464924192751849273249865761341412961600243528301525154033206984420337
556616976173553343451538112761782303041577387865173576020145899030695462069065
351711300188076000520650939052152800389076847099469601808412030997450411866718
328611865490103856545153616005988380044924945256557877614064389782665774206832
411967449751346615962876586946213686698711643631275635335149528646004828057278
004190419870510081657707240659137583988000520343636253480334898690385149103585
368570634580095882712702849300109928689359987147468849238926334734854245412688
715146867566169170681967207525691336022089090287580952299456594764150612460738
255998885216745605996111036014063528612604711748023315552566811553927437855567
821790569366659293375152951940933326091846631854064965008359063918058917234835
442292916243240337502602422687344751988706321787037398353551039243691555255875
134457059916448337962447184707856118983477102806472004318967382079920904030839
766456322612605341318596676907721184490942249405667438216006927617235153525030
064785515025246914076042976716405628085257548207585396903651441272704392969943
543335542786305484005916868738198135037240551024131805174572887029028088900945
291583585387057809581864534993631118974153781849387603620472442672135255961972
208917645443775501912886736672085158006159806869798589689978095240193893776673
669939084760586617405535417096185087097591926431304283589554387992282333626690
065420444365082678634670456938640551165035774843379086732945840022593865759213
129260990724734135051952915555996034482405687168097119486489531430358460283122
642466930767626264687114432265211767329963781209445839023649810458309943551504
463508766501226047939441972191732626302477188434488279340375218359882883614235
148936985965730840576683171954315773729991983345198933493450031507921294521787
309312597932496739256190155259719148524028125496609987434211444047746055304876
474832032350155395468783544850522032053755596629519697359115070371004247610373
773765816931170121558488082530228949284872941174054324465711538587531947432425
261498389632923471726393502734990134247026932599049778970450822200690589326316
848336703688167279661283316060673882330154485559722831554953152790512944632946
735062955119167184229698846592671871061956988599224929488766360362711791273491
242638869726650876864081880699049446615099393830619711646796113001921819683024
369590515257225336237992051139086240802237202937259314304211737816439025200542
055318136632596576723199340502998514457667643516144415500155722104031898805440
653709255083200892365218872640883469778246338301021065367088378738509704040810
118194328693552303379694930125516918347623107028885523667957806037022051779505
028008670887159641266810562570756698846186139861592744870102722702811328809624
868102930465135820109650384254656351993944445309647709325824379714345859125506
877130290546602279165189060434774131962949884052040496677412243316096744620693
327264458120474035506817919611461758654045587703663849715440195969689615628960
733445735296123199491057564399722505169785378990925113661431487432487733678949
019352988522538094902522590394751351923716447424611528634448882020511514348257
497663043958452914278076401286422973635083861449802065395422055989305365344258
406290277926136430522703183045682790609399758778618475976902240574620411468420
789028389415679841570668448757332730136446415250393630492420794579690598982563
634794371786030627807571427540544467716211510439649619615894455342087862656397
318775137299062662894178988094097600508351513266859789769143450230405235877948
795703108619774510513482618808266634220911430043634780135372644293596024532610
984706096099075879394304168910801183438306189422268343209329596285393195315550
071036782813675559163239612697474390396580444802778813013814637945415280117737
606393331747556123925865982509562883171017504543694725101038457391872858194293
569737146079314281231269511124082599342793333822398383287610844889944177275061
276589196398144868452633037548205588433810339011632429729270198883661339140821
828082433113732673457903301183426283393331797507107656636715689233784438799106
754768243983336732261591094015393469562139536566565776453479816009954691934306
373112756944918730123928293059783643062895186209981463422826315903634488313032
831853876784002124661481417309822646974566572851980596186466222535610586501060
559389354250986623486636672918500396484429115992054143925656145321307762701431
109862585265745263460848266175202942100591718317478075708259661506302326093932
413832915555816215547105569544654419443851662259659900478861906607307315711016
958544747987869384398340404111546329077619079464671118673247160362213345007884
829040611627439276302939341999583743108136532968942657182539530873951450191820
935056069335704163143611513476045912657685600322963842457985498317907126091011
797318835314314701692825723944005383968675370154720935387808230212632056793375
926964558766056261071327777661355290321651670368059861572614056061488427629824
192828246006011960072285445047286228865219954762507436349751805180986235154915
226834386051053006199966404486323469711312437376372424449414920445067932761767
679125806576866861237969161787243497138590205823571609427520177842766709146463
931539594188401796843318630461527843189738406253298660145756988009167644881508
215285259433998864854474587186924340753953328502139615847864693367792757871489
334294260541692283370771062100289635636419880551887157493162875686211754244619
227819039025740912688153913374136780795364905670678695088313722635759543577546
141771415471902552505807846123397894719299325008588235198590011532298380379162
425791005356111440356331256325762107359190362635732542558624100131583979524754
316910639932572932704180665214512726135049044612440766293011804330394319279334
810771916902515015037158492688824885861937895558287133963500921684628186124273
474060259399605688305532735318473816804027047283330280043143429185793369240273
355779706173195060993551811727692743520160416876902904340402564908668603762472
042907215970259047724490235316306370606991938770244989659716703034571246995914
394501207237452604619273074148348554404473259576213397312997544762714495082959
312544595205212262078197542384527016059190673289589876424310783621937224230258
950434962433395477704784381079837711221394327892603565116857119214328127573413
402333253237221049254791185017480310218989543061451931033189737257000890152990
004838268071419086899819918106727366542493889477200541334737148591196773655501
134878593905921337885561809629414995199061663656519267339522270025748280644810
98321959914380446

-----------------------------------------------------------------------------
Zeta(3) or Apery constant to 2000 places.
1.2020569031595942853997381615114499907649862923404988817922715553418382057863
130901864558736093352581461991577952607194184919959986732832137763968372079001
614539417829493600667191915755222424942439615639096641032911590957809655146512
799184051057152559880154371097811020398275325667876035223369849416618110570147
157786394997375237852779370309560257018531827900030765471075630488433208697115
737423807934450316076253177145354444118311781822497185263570918244899879620350
833575617202260339378587032813126780799005417734869115253706562370574409662217
129026273207323614922429130405285553723410330775777980642420243048828152100091
460265382206962715520208227433500101529480119869011762595167636699817183557523
488070371955574234729408359520886166620257285375581307928258648728217370556619
689895266201877681062920081779233813587682842641243243148028217367450672069350
762689530434593937503296636377575062473323992348288310773390527680200757984356
793711505090050273660471140085335034364672248565315181177661810922279191022488
396800266606568705190627597387735357444478775379164142738132256957319602018748
847471046993365661400806930325618537188600727185359482884788624504185554640857
155630071250902713863468937416826654665772926111718246036305660465300475221703
265136391058698857884245041340007617472791371842774108750867905018896539635695
864308196137299023274934970241622645433923929267278367865571555817773966377191
281418224664126866345281105514013167325366841827929537266050341518527048802890
268315833479592038755984988617867005963731015727172000114334767351541882552524
663262972025386614259375933490112495445188844587988365323760500686216425928461
880113716666635035656010025131275200124346538178852251664505673955057386315263
765954302814622423017747501167684457149670488034402130730241278731540290425115
091994087834862014280140407162144654788748177582604206667340250532107702583018
381329938669733199458406232903960570319092726406838808560840747389568335052094
151491733048363304771434582553921221820451656004278

-----------------------------------------------------------------------------
Zeta(4) or Pi**4/90 to 10000 places.

1.0823232337111381915160036965411679027747509519187269076829762154441206161869
688465569096359416999172329908139080427424145840715745700453492820035147162192
070877834809108370293261887348261752736042355062193737506171117453492968677507
330760668693411890586283379527951203344958904688626269482208350329836321490205
321239557248466462255011566604558826867876535044954351371974951488631328974725
885751455324761892324749088343183216559962899648054020498855660906710813145472
438251775250469502552514132207698095596477686277529297400362468833633531227758
668098332337740208149807142410957545732327968829227632494222596768214873164210
130083383048578880296049867785925835977212171325665188800281027638581269931387
545523778137133635462656850510299889287023951692086070339616088259169901546588
829589456014838452545931150701738279902071569195340951585273588301542879433793
746390084034611646372042627028266824368992792302461651111441102686560899069254
234916963890635275530295265490294558420276762570241101886780740855794520052537
619233699975159280962799577357881560429246227564596438489387074199902375906750
607371161632214547578450995034633065293054464210084330411049428823987182779472
210717957162013973497506273354919182726537315801429070644112238984462888749346
123621785684422095337765665063123393418309331590366489436671353950052108795316
627967870123212606511753317383291994351943063709445458703075841875966315880470
976209778331910566278373749222678619336932243629212937680214776276988227275161
168195674581326992671514767615743996645318562807522634712756683205732676873043
690872655267444910730887471434771768813238590527416830613773207408761753459968
131671551610024798988217912278432609940552842489853554822285939017206671602845
270243345579339355738139745080825577027485254916354927551049265465602698556315
339949578895606778338186500146965223230533534669952369976343379670422876496446
956754049652503997706078434631929556382871737860047504977021567642914163208355
134805792566068258029946197161166626296206873985365333182877235965645025559136
257276543969869988263262252546952360187873782368673354821387004235792861595706
529762634076158431188855993259791932713520010274020791811758790707479416547489
106183607617773678547349113072916281762908896372877205362179853486880174699470
107383795934201535381924332932601995742371149445554137644649833874733600917286
939054275258964969221373438159681513527365152368160859347449314348075900345248
022371248340709893964541153866142119927965830143459708005575640219601213663806
527389078736847155824812085033248336785809277461345965241697530456781539113888
521821359486098822884971197488419528446391075495066353252263979396256780568023
412994361020882530460315369615010615907640388449002875649111956761004893444788
262667636891482868798355412860868083881677258693652904007497955167213959088437
014275117694129854270050771940027704855368527442476609058963047355537726975352
606857483276704063949198474837601584854183457622643688691068255182852613072480
280116483639755956169869839328933485751218575154980335069046376361009455721878
935484849344137476171947848917049046209355129479618398660809902190592737601620
169040890810646508898460380317543721534054169065012444317105131721986309172378
537608940683232656099437288581269590045631303620508424266923250471846659288150
681335483184307726970195422317477929132443869159416282481643778054583836828018
127995153507811603759817449284836263090201220997756637752944272923039050840588
526655478542632456194649397920321886086276148959318204123715442572261905064051
748829412167006730489244277871224198449924807997912473097994195519502920052843
575035251296805230980217455834950259915190120471703736893969162481741729962970
670053192448016331645644501333390769048925825620404903143689985910667455231442
887398142474022114676678064465714785776073328090060077734808126840056071518617
356668107821035230382280947820153398762704481691191400260576150041157413347576
742879641915584829784627645250575918482908054004437109628179124887284065326411
054627384875022666544898574091494201280709248084335891061347290176511364024513
441423900902897372088502949293584365924754300090827093088621890103369743277336
769064588923034773421546535544407594434329898141722985633954282052510164526233
913719661570085582247239392544908888300091769494179793676834579998974491241613
469362079826316698854209890820308211234488011997987527096062316655884478678345
562907661212199923383118292350529211189664912273995319676683701427948310475897
577153839228003557654505766139993562625901573533844672016287860983673445659563
909289685025191059262276701583611665557106040610390094656438320903099187970507
379881452327258878189833383204525240037165905438076399538113828388453955777580
515494214492670669082127457130974283485983092914088726492118497432266371226824
875948389614981716856752378501606845324359404776151498977944763288087490189431
567980871984467577345685128211910689436861931973467442977900191686105709960704
247139640024918766861671955618581917473309178753765735212385607298227106480106
537149722093254436283421847474576282179725315520039011067876317406749486747639
347982896766183337576896202860401071552989371016008309060190992412809088614441
646654105893505762407387440923795927678677314054708985208520611435535408016654
229067285430682554469717607581231080604156967903901761908112936876560748896341
920294370713216767793388638761930280085425060461906222473604145351687985760903
215643034996835766181131997746531597187699717610564527952652882093884115991577
451698549942629678093993472726354309939190357942645720968543107431197468809980
627437570862144295710924616576802276874075744221432380876748250155470605742281
378287902640822542634919080798704990776680130020788878239540901626695762220205
937426713516690116748889182776348228040546300641164787404406846000410779257654
566066199341299327168946834456817887776921829442716946532432872451403061272891
309864365066980489383509916286022409613910077568526109009803801697563225410881
950459510228002933350489023164173650662870691434106648948761237364686556492267
838944877415623554815992549230522653604466802762074124648518031631580208556771
023187378297340776463563573833082175333630258288793190155782640058468749652457
996150129530418758305161376339184708678280858577350937785605379534065498551497
866972412114339340547123309156099235223158976599543868646554283116896055533026
588884424173206255489058108345076191975111853834936211623838845586846324355871
323198354712042834149285939765092379690107996960029174023107490552191156034580
023577266313786111862027439018455578840193761925743604084791535372635361671233
705765231285599621878967773929195780141337502382328586041449205853926138699511
458137888925786064551305863378072686216164952071092952231512001980819891830651
371894098521725499101593176294117316758576093583284981504936288147974143797550
245657411539099817339013176215794226154818161277415926854913299575201061876642
862121797806780983628315682131365868377374828850129965203712962056795825599709
646981304892530503728127191808151672132660546730813201527177759365397356048516
828894770528471817758009006338903221582017792829474029923652019778993738914921
117470039490882761095481772320518445627150665366862783244999904054017729572390
732538607432181489933039851528943669643773528976332474171608630941090859597363
114783188312239778609992972517596337788985112084623349508280843281200428058607
718811414621972965504794766101978529681596048441475728869508540567316768114348
889138961520300788934642309259598715273033783291534465746894737449532079599303
618266368031324230774053892443623086218378228126439958968424445897200361459364
463623172532572596661183895027638353423444389643114391115334388645902941449500
732571763151162628164649829973364651802997793189287950022681105360372194839533
016176403858484554564108507065133121480345105324438099040318190955849197497061
904745526328509255928560324798988508452008254659458580751965574079575664172151
429283457530334433832247339205550340081934271407875523588239038928688231031116
685774070902353399406220553935353359705991585294396065635832205296490274719954
954545635571104827608342293449851043206010425769596017548513029751046669457729
911002771224625850764192851256486039760777545031797863829805430625454242197192
650845549179102957624514082855678903328840344325001075325771190441903524679102
916635927653682510365850466208370557558260830505848475444490942055517025514061
236295517217126857903003771312835776356693150237134797866633953858174574710649
136579953232455487107124841923945710055281481281172026560468060409693818161392
120593310600816179615501558311898550270948482669262547402586210264886603224423
444239787177354946164616170997318446945587112298937701381742794064800317168436
849421744820503433814670428017724395345703671119167117585728943065604163190547
839533264443118121740628530720157344406090463915631451724267831928308834119239
736248383292471178282470624518063900179921030228782015138996734766295351374675
554673622002323542920078983611532099684121749186225665063965240769129740102540
785260559694885346373325632780322499928187657550733044222657648029891583042976
473511950585855477705357784614191197986186803349909050399051684277447168462645
827309613058038524324779844038106483284405352269783143279061442328225443426863
267646966313161361597434286041079509870675820652906643205446812078165087198608
410452351966497715341000213896226639542237898664203168559646819644328848841845
270367785915074494145826117421318834712891456565891957492382282189903366592406
972810028353566198885128802997885995081975922287980943864508926875597853178261
854102911427737034499520706208814113273004255217324799402676464990116279901559
293602974751606325065922379687952954853542015714664761686279842547745463992391
558076861160339246996760129883338283079538922436438622983346118907620620672087
053503624129784260426616778509719545312833367099778911819532601228794907920967
614092904418446876172334172096949688116212054509841294897097252924033729833667
006043967308172363020394063078521448894823418466897974356016880504076300639363
41762674388635891

-----------------------------------------------------------------------------
Zeta(5), the sum(1/n**5,n=1..infinity) to 512 digits.

1.036927755143369926331365486457034168057080919501912811974192677
9038035897862814845600431065571333363796203414665566090428009617
7915597084183511072180087644866286337180353598363962365128888981
3352767752398275032022436845766444665958115993917977745039244643
9196666159664016205325205021519226713512567859748692860197447984
3200672681297530919900774656558601526573730037561532683149897971
9350398378581319922884886425335104251602510849904346402941172432
7576341508162332245618649927144272264614113007580868316916497918

-----------------------------------------------------------------------------
Zeta(7) to 512 places : sum(1/n**7,n=1..infinity);

1.008349277381922826839797549849796759599863560565238706417283136
5716014783173557353460969689138513239689614536514910748872867774
1984033544031579830103398456212106946358524390658335396467699756
7696691427804314333947495215378902800259045551979353108370084210
7329399046107085641235605890622599776098694754076320000481632951
2586769250630734413632555601360305007373302413187037951026624779
3954650225467042015510405582224239250510868837727077426002177100
0195455778989836046745406121952650765461161356548679150080858554

-----------------------------------------------------------------------------
Zeta(9) or sum(1/n**9,n=1..infinity);
1.002008392826082214417852769232412060485605851394888756548596615
9097850533902583989503930691271695861574086047658470602614253739
7072243015306913249876425109092948687676545396979415407826022964
1544836250668629056707364521601531424421326337598815558052591454
0848901539527747456133451028740613274660692763390016294270864220
1123162209241265753326205462293215454665179945038662778223564776
1660330281492364570399301119383985017167926002064923069795850945
8457966548540026945118759481561430375776154443343398399851419383

-----------------------------------------------------------------------------
This number, the Product[Cos[Pi/n], {n,3,infinity}]
is the limit of an interesting figure in geometry.:
If we take a circle, inscribe a triangle, then incribe another circle
inside the triangle, then inscribe a square inside the inner circle,
then inscribe another circle inside the square, then inscribe a pentagon...

The radius of this figure (the number of sides of the polygon increase
with every step:triangle 3, square 4, pentagon 5, ...) approaches a
limit: Product[Cos[Pi/n], {n,3,infinity}]
Is there any way to get an analytic solution to this? Like this
would be the square root of Pi or some combination of radicals
and irrational numbers? Anyway, Thanks,
Mounitra Chatterji
mounitra@seas.ucla.edu

mentioned in december 1995.
By Mounitra Chatterji

.1149420448532962007010401574695987428307953372008635168440233965;

maple routine --> product(cos(Pi/n),n=3..infinity);evalf(",64);
------------------------------------------------------------------

The request was sent by achim flammenkamp on Tue Feb 27 09:05:13 PST 1996
The email address is: achim@mathematik.uni-jena.de
The number is 1.60140224354988761393325 (to 24 digits of precision).

-int(sqrt(x)/log(1-x),x=0..1);
-------------------------------------------------------------------
.283265121310307732587685540450858868452123075913479495609303244760289207466703551200728343246718266
1721794706326872389237418265273196389116929121819750888062495294277256191719424273967384545908106616
5124702322513598413388920213387535350692362866707758376138858482266928332718882186473891252470626193
1134162075403008037881499615240658150936661712754874529120769279078826146925069339158824377250780006
81691683658433538480533518043146405030754456294577975558177142447872562829157

There is a pattern in the binary expansion of this number.

The request was sent by B.J. Mares on Sun Dec 3 15:20:18 PST 1995
The email address is: bjmares@teleport.com
-------------------------------------------------------
The request was sent by Joe Keane on Sun Sep 10 05:02:26 PDT 1995
The email address is: jgk@netcom.com

The number to be tested is:
1.38432969165678691636600070469187275993602894672280031682863878069088210808356345

The number of correct digits in the number:
79

The hints given by the user:

It's log((3+sqrt(7))/sqrt(2)) or 1/2*arccosh(8).

--------------------------------------------------------
The request was sent by (Mr.) B.J. Mares on Sat Dec 9 19:10:27 PST 1995
The email address is: bjmares@teleport.com

The number to be tested is:
.86224012586805457155779028324939457856576474276829909451607121455730674059051645804203844143861813$
451257229030330958513908111490904372705631904836799517334609935566864203581911199877725969528883243$

Another binary pattern.

---------------------------------------------------------

The request was sent by Jon Borwein on Sun Nov 5 06:09:28 GMT 1995
The email address is: jborwein@cecm.sfu.ca

The number to be tested is:
.01118680003287710787004681

The number of correct digits in the number:
20

The test(s) to be performed on the number:
algebraic
--------------------------------------------------------
1.456791031046907

The number of correct digits in the number:
16

The test(s) to be performed on the number:
algebraic
gamma_multiplicative
gamma_additve
zeta_multiplicative
zeta_additive
psi_digamma
linear_dependence_salvage

The hints given by the user:

p(0)=1
q(0)=2

p(i+1)=sqrt(p(i)*q(i)) i = 0,1,2,..
q(i+1)=(p(i) + q(i))/2 i = 0,1,2,..

x = lim p(i) = lim q(i)
i->+inf i->+inf

--------------------------------------------------------

The request was sent by Olivier Gerard on Mon Jan 29 18:48:42 PST 1996
The email address is: quadrature@onco.techlink.fr

The number to be tested is:
1.062550805496255938

This number arises in the study of generalized Zeta functions
on non associative sets.

--------------------------------------------------------
The request was sent by Michael Mossinghoff on Fri Feb 9 14:40:28 PST 1996
The email address is: mjm@math.appstate.edu

The number to be tested is:
1.296210659593309 (see below for 2500 digits of it).

As I mentioned in the original note, it would be interesting to see if this
number satisfies a simple polynomial of degree > 34. The simplest
polynomial I know of that it satisfies is

x^38-x^36-x^34-x^29+x^28-x^24-x^14+x^10-x^9-x^4-x^2+1

I found this during a search for polynomials with height 1, degree 38, and
Mahler measure < 1.3.

I also have a second new Salem number that would be interesting to try.

Thanks for running this!

Best regards,

Mike Mossinghoff
mjm@math.appstate.edu

1.2962106595933092168517831791253754042307237363926176836463419715400357507663\
555372700460810162259842255138960885075885472138523375229647035948031308222869\
213377761985420998401465270339786283142588526265385851765349326219909024384324\
298668143261669279113959085262729367911041451897621484638159134108808507417558\
371227480609429111967509190900525542468572422201267290352457473788303514632978\
531591219560940258062757424400763572149784569551257493407108061275808255266204\
988526404732083078237046586577078037338486088388181584983281574252897177808263\
147692481736785688370028996889741999268557158363474402864561998038209817582814\
010732290535268946721928114002527443568020359790313185377702725896115435126307\
841519785171242185997657977732689357703555840184684554577244752237497568339160\
938205575175811976414747122955198011255949965359970687280700475477368518212756\
924749820065045209604606889253335548989681523027453599219856774850675170030081\
340461412329460883636590018878175768282781839837697211776636498168350816554156\
904601023147786817236407289883278093415918634119620218433047846657184261144649\
040715513536648841284787099601551612909626813632800691067564404454541790010887\
679088108728482285977923782153457884089162309486388513634809308430291906873755\
353865787785568433558148544650806363798445573460997103012214477139122206697676\
151378710063572151250547043624062114013563819037462333697027524356258777528864\
271328965733484293667236211401267719087175146826163754038706366216877272628132\
296182344392845125506127123945469182368918766036231606918375224969603018840277\
778903237698826183111400261578682603995590568903906955569848314084496482503972\
906016618276547328327517227379822958377122743985938689837061722495995392321936\
345285971817821600170724492417762482659737742843585759061520292400466743607983\
593438732628413114256276767063139352552076489085606199932942061150333621663624\
667294211959583161911171198313494502505440901133068426838051637173543721800267\
607050254597479936347302850855318828765200608121163125879643065717811879123723\
939826702878343201235748915166745912187493987556824139288848294746007488299743\
663817162198495190194616103659925459932420514340386336983265209362290719538034\
616103846861918706369114431911997889483119661422295458652413962075819025018423\
406629086461013112957825351840936858715307617702746177132615020866202346765384\
189199689332174745118809280247719860161398327812075021357273956644275172873038\
687900608249173662145494837168975704911668609774430992557238265593517876057742\
2513

-------------------------------------------------------------

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.php
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

-----------------------------------------------------------------------------
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 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

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