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Title: Values of the Rogers–Ramanujan continued fraction $R(e^{-\pi\sqrt r})$-Definition: For a positive integer $r$, set $q=e^{-\pi\sqrt r}$. The row stores the- Rogers–Ramanujan continued fraction $R(q)=\cfrac{q^{1/5}}{1+\cfrac{q}{1+\cfrac{q^2}{1+\cfrac{q^3}{1+\cdots}}}}$,- with $q^{1/5}$ taken as the positive real fifth root CITE{MathWorldRogersRamanujan}.+Definition: The Rogers–Ramanujan continued fraction is $R(q)=\cfrac{q^{1/5}}{1+\cfrac{q}{1+\cfrac{q^2}{1+\cfrac{q^3}{1+\cdots}}}}$,+ with $q^{1/5}$ the positive real fifth root CITE{MathWorldRogersRamanujan}. Each+ entry is $R(q)$ at $q=e^{-\pi\sqrt r}$ for a positive integer $r$. Parameters: r: type: Z- title: positive integer+ title: the parameter in the nome $e^{-\pi\sqrt r}$ display: $r$ constraints: positive integer Comments:- comment-cm-point: The CM point is $\tau=i\sqrt r$, so the family nome is $q=e^{\pi- i\tau}=e^{-\pi\sqrt r}$. The order is $\mathbb{Z}[\sqrt{-r}]$, of discriminant- $-4r$.- comment-convergence: For these rows $0<q<1$, so the continued fraction converges- to its value inside the unit disk CITE{MathWorldRogersRamanujan}.+ comment-cm-point: With $z=i\sqrt r/2$, $q=e^{2\pi i z}=e^{-\pi\sqrt r}$. This is+ also the nome $e^{\pi i\tau}$ for $\tau=i\sqrt r$, the convention used for HREF{T295}[the+ singular moduli $k_r$].+ comment-convergence: The continued fraction converges for $|q|<1$, and therefore+ at $q=e^{-\pi\sqrt r}$ for each positive integer $r$ CITE{MathWorldRogersRamanujan}. Formulas: formula-product: $R(q)=q^{1/5}\prod_{k=0}^{\infty} \frac{(1-q^{5k+1})(1-q^{5k+4})}
- - r Numbers:-- params:- r: '1'- number: '0.5114284554037035192946330135425788104157543814174665124187982080507562021874566144382624867458127297'-- params:- r: '2'- number: '0.4064605812995065920896182374103892775964745719514374630420405047120796284213807241937612229976144283'-- params:- r: '3'- number: '0.3353422604252457377634254037231980060911401804469839137514494403981197158034830427585877742249190029'-- params:- r: '4'- number: '0.2840790438404122960282918323931261690910880884457375827591626661550458773514845537303784177522316259'-- params:- r: '5'- number: '0.2451580725478960661742616314868277784988095784262814050649070560543180476935191058932106262375090517'-- params:- r: '6'- number: '0.2144851757810122094843330484296364262159774762955522817042235012528848654272534330601769719470861147'-- params:- r: '7'- number: '0.1896414433626258810635833635358665334551168078243071176454224940962761159614816852362060618893602495'-- params:- r: '8'- number: '0.1690955211413606647324925685630614029229222821013405754306635661791216675199939804389706354456199562'-- params:- r: '9'- number: '0.1518235498934964612889308227856287827262113648732169136778450380297056040232555701504761174628340209'-- params:- r: '10'- number: '0.1371107725802946953936999666911150326625769604237964781548830393212795425045014729725418859154115993'-- params:- r: '11'- number: '0.1244406155944850389345146658271114198662517270509441941158684253296282940477084817669991642753205119'-- params:- r: '12'- number: '0.1134290196256987498645032183805072562556239227331849802867129861912415464701439372773742128680384351'-- params:- r: '13'- number: '0.1037836618433571538341555035953966017744924423011021010075793509584941643885994770154726600613392364'-- params:- r: '14'- number: '0.09527760187294194811057419769441411864911856350813810626094230031141715959738319396044541438942549896'-- params:- r: '15'- number: '0.08773166335795842144575528494303693200453342602152481475771891720081893540895335450675022950524175058'-- params:- r: '16'- number: '0.08100230967515765130997208783934959185511642003879960863612336429419885436704082489739142811537929868'-- params:- r: '17'- number: '0.07497308942488644807926852923533727633291941276025841301502134273459658306520694646834856998459361529'-- params:- r: '18'- number: '0.06954846786635897000329366473882390663577399276122517584028519742150830572432593996832432152217392771'-- params:- r: '19'- number: '0.06464929303543405613383876419120455844244789022917027481079190067446697971834722547514327827146654697'-- params:- r: '20'- number: '0.06020940656245182043572967581957492337261937064631186642499705561396077550229157404778396819126206185'-- params:- r: '21'- number: '0.05617307177318694688930036962423089132354626298667960771464204218178390582565297196694090372182654392'-- params:- r: '22'- number: '0.05249299550482749878731224588165945228459438465231388342542003021631429046940293099000172290699652041'-- params:- r: '23'- number: '0.04912878799236763704821133969954768737081142248997483336190931287436007905879859617065661661298093385'-- params:- r: '24'- number: '0.04604575055945772409276228878245288009776200402395567064240712571631505509867404874199779446014222458'-- params:- r: '25'- number: '0.04321391175136109512627636216954364995889851512776411554505360353128650494640427090891341122261566302'-- params:- r: '26'- number: '0.04060725396296401859427765216994914400133354386213344294351559757769546227674620851397991986412863779'-- params:- r: '27'- number: '0.03820308769222036555134883174771449152228051843062227742572742133126662299133392586827759043052829993'-- params:- r: '28'- number: '0.03598154131985774060641670212085857014368294363489152763389369609684884522429079370047810986059248572'-- params:- r: '29'- number: '0.03392514211317227066765247260560945671845402012350759430353072180434769283021960011019256395228973433'-- params:- r: '30'- number: '0.03201846986591873259999054740086854038034452707341757754216517634514643809178991528684235414193265484'-- params:- r: '31'- number: '0.03024786882183901665678717265455018739530499021683733437850934768831981376981110038857709639660349639'-- params:- r: '32'- number: '0.02860120670208507364013960637690325366829664221211978204029123852437190033941455716387394332381295160'-- params:- r: '33'- number: '0.02706767205672242200644632679203631962024662262896583947614755260235238698892202595428049987573263313'-- params:- r: '34'- number: '0.02563760299248542922101969328881840547662414834349555033106951891072978233587609357559841690383932323'-- params:- r: '35'- number: '0.02430234173934742380730549000286557919335623394065430083484969975472434505900932748471103481820729000'-- params:- r: '36'- number: 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'0.01624191325380269481427343585263721595570304728258997057953987249639683983679064453455985544794384159'-- params:- r: '44'- number: '0.01548639114551371674380032861899938540479217138191482842612082842442716599243636875752246744225608974'-- params:- r: '45'- number: '0.01477396395956972524093501163260669563731236670637551617483454203223530214517163650250676968330273387'-- params:- r: '46'- number: '0.01410164798020127463171929710751678380722608771953588948679146635113216515707035718713739224572493269'-- params:- r: '47'- number: '0.01346670648235895975977717233753343896367529157155522203193154413964194432398085423261487583274917913'-- params:- r: '48'- number: '0.01286662570781773586826611445305896916686201531037480027163653153368491613268053602851783526750317907'-- params:- r: '49'- number: '0.01229909353935142020918119156279785760635650940579284081680187697643320204270519926300792072000617617'-- params:- r: '50'- number: '0.01176198052874133656912513293680090347929300124675486872622133226365881700809279913374219017975613560'-- params:- r: '51'- number: '0.01125332298356321718384754530526296690268359147565052051805318888497524187566348046568692866647799391'-- params:- r: '52'- number: '0.01077130785910654160191741030803559460362780683829421756895620847819590084231363183069866734175360761'-- params:- r: '53'- number: '0.01031425923676357922767245726190433202978202898508560891838723798350882423872629335781246274724564335'-- params:- r: '54'- number: '0.009880626199872929032494132521680086596818931658701419579212978365321932296626825855480721783433528406'-- params:- r: '55'- number: '0.009468971943204775091309597583417675594478191509427808098163230680945783578409630072307622183503246127'-- params:- r: '56'- number: '0.009077963973761708067964832224261400676735632729315044620911433957319862971615473540904202566070751731'-- params:- r: '57'- number: '0.008706365278939338509869076027669745653080688925865514450530672168784354549262593312827351516861686084'-- params:- r: '58'- number: '0.008353026353839665373341926259801238868613037212722307753996735276186643653740242614711730185859655032'-- params:- r: '59'- number: '0.008016877993066722358263143619398248148600075056915992495585866461616848612014774599929462694273526695'-- params:- r: '60'- number: '0.007696924763998489935244432791359697099576804385999482162951114672357172959254382392064610980084884956'-- params:- r: '61'- number: '0.007392239088604796340879025821925511547587360804120954593106194417018407461896725983049566744007213293'-- params:- r: '62'- number: '0.007101955869604765430018516679789061061389252748741759241215029928917039373435797009174766542652595445'-- params:- r: '63'- number: '0.006825267604327826629905430173210763696787729202186542137719539205593528115572487136057289573096134389'-- params:- r: '64'- number: '0.006561419936226272588622208626907442508981648788371991612737363818672391349702078731653179389751544573'-- params:- r: '65'- number: '0.006309707599725519228254231013632469475235872119254620048224732169980991581960757888095977960752493279'-- params:- r: '66'- number: '0.006069470719109503946037580961874862169700402033217543986738771787307662136702836765134681746653812256'-- params:- r: '67'- number: '0.005840091426523820851669067342401763285906702048266559981848716888672635513028725891290295007647928055'-- params:- r: '68'- number: '0.005620990768023846478632801640164039744860649435433140913331370363490102210670584720842103347339642725'-- params:- r: '69'- number: '0.005411625869972161142024765894387070804393981338858929264451168336614713634892122424697746654544994925'-- params:- r: '70'- number: '0.005211487341061253297339512038765109978292594992296561044587034385803306658142708236134433206277746515'-- params:- r: '71'- number: '0.005020096887857050990042254729621181108529723164302329137287635108431272357320770752622152657713876559'-- params:- r: '72'- number: '0.004837005124071909802808507124629690730773706896327513090970772075419756327157026681353052799463088640'-- params:- r: '73'- number: '0.004661789555821530471842804269292651003984566467381036840377804716665917926207014975149157805103565013'-- params:- r: '74'- number: '0.004494052726932661389529492083028200705356682528794966659767017412874785885328779069347305557073606534'-- params:- r: '75'- number: '0.004333420509976507282795463318386785502826150449864450280145590924876028478703520332111017636117633903'-- params:- r: '76'- number: '0.004179540530131712225820835748057895347481979430152011609445280964386274819963989313313697327093802437'-- params:- r: '77'- number: '0.004032080710252437238947428554645958079463116295027343479938372322000375694488603773208781008079040918'-- params:- r: '78'- number: '0.003890727926650348569691886877006682288556188348698372760772885446573715222764901445599002897335507282'-- params:- r: '79'- number: '0.003755186766110736258110700748970760228937814830462196062951709471077191764811105336803599613639053070'-- params:- r: '80'- number: '0.003625178375566831550333363087508202624991486517011685665963580763885255101114378341440261772740695507'-- params:- r: '81'- number: '0.003500439396665193737363193991764638037942477992470799710811739753407351182593205790210433038960852171'-- params:- r: '82'- number: '0.003380720978179721259574541684923828704043152882085247705255326505700034033807459781548126108471766694'-- params:- r: '83'- number: '0.003265787859881973916002260429075346899879081619471748586593313547936132045231471842630009764724603278'-- params:- r: '84'- number: '0.003155417522059457924935334278178212847377193569129481809527614431497849508446858739557240309463500356'-- params:- r: '85'- number: '0.003049399395398685235375939359539334867238371803357185137508884323536099017057067802917984478517487631'-- params:- r: '86'- number: '0.002947534126422645995452140040991326216954681984210647664475644117501777565577052519264497180439533556'-- params:- r: '87'- number: '0.002849632894098528459235569271615533027254294458383162539062675686384096031917526410127303546004865378'-- params:- r: '88'- number: '0.002755516773616110295795565977551596796882759524052086391963216497003910294113797099068585968060059938'-- params:- r: '89'- number: '0.002665016143684667904343106566107072202793240007475801506644948402518151643354020652910910454149370861'-- params:- r: '90'- number: '0.002577970134010430881887070115645458570890547962012608241047495225204301899218086827207185523465199077'-- params:- r: '91'- number: '0.002494226109901022551322461138661229127574357228686939437109259078174938471563618190273376903834326800'-- params:- r: '92'- number: '0.002413639191201055638964307321000003447720808580595294639675794438145118452237038719688563013107886580'-- params:- r: '93'- number: '0.002336071802996830044880115417948755892475782314672773914517052848155038101784731454215487616482937789'-- params:- r: '94'- number: '0.002261393255740338343917698408586486409061326308533800830936907963649513772553885239930001470745074387'-- params:- r: '95'- number: '0.002189479352635687703071990005672102261562217305990366596245748418342718077648521697401407934338999078'-- params:- r: '96'- number: '0.002120212022306521916197924207005679925446370077770809007393236301678885890887648489880659594957477709'-- params:- r: '97'- number: '0.002053478974922793674109658353869029489818579297190141279660920092209877754888698168840212880238056189'-- params:- r: '98'- number: '0.001989173380110830488772222582676916047924689577121231864886653295542617528147730893253954213380944438'-- params:- r: '99'- number: '0.001927193565103430611263820060957977387723617964156479622773335505217959186502287824605689934946594523'-- params:- r: '100'- number: '0.001867442731707946402918382775014798517045926243423716872473730757485005391632170538360673564420818617'+ '1': '0.5114284554037035192946330135425788104157543814174665124187982080507562021874566144382624867458127297'+ '2': '0.4064605812995065920896182374103892775964745719514374630420405047120796284213807241937612229976144283'+ '3': '0.3353422604252457377634254037231980060911401804469839137514494403981197158034830427585877742249190029'+ '4':+ number: '0.2840790438404122960282918323931261690910880884457375827591626661550458773514845537303784177522316259'+ comment: $R(e^{-2\pi})=\sqrt{(5+\sqrt5)/2}-\varphi$, where $\varphi=(1+\sqrt5)/2$.+ '5': '0.2451580725478960661742616314868277784988095784262814050649070560543180476935191058932106262375090517'+ '6': '0.2144851757810122094843330484296364262159774762955522817042235012528848654272534330601769719470861147'+ '7': '0.1896414433626258810635833635358665334551168078243071176454224940962761159614816852362060618893602495'+ '8': '0.1690955211413606647324925685630614029229222821013405754306635661791216675199939804389706354456199562'+ '9': '0.1518235498934964612889308227856287827262113648732169136778450380297056040232555701504761174628340209'+ '10': '0.1371107725802946953936999666911150326625769604237964781548830393212795425045014729725418859154115993'+ '11': '0.1244406155944850389345146658271114198662517270509441941158684253296282940477084817669991642753205119'+ '12': '0.1134290196256987498645032183805072562556239227331849802867129861912415464701439372773742128680384351'+ '13': '0.1037836618433571538341555035953966017744924423011021010075793509584941643885994770154726600613392364'+ '14': '0.09527760187294194811057419769441411864911856350813810626094230031141715959738319396044541438942549896'+ '15': '0.08773166335795842144575528494303693200453342602152481475771891720081893540895335450675022950524175058'+ '16': '0.08100230967515765130997208783934959185511642003879960863612336429419885436704082489739142811537929868'+ '17': '0.07497308942488644807926852923533727633291941276025841301502134273459658306520694646834856998459361529'+ '18': '0.06954846786635897000329366473882390663577399276122517584028519742150830572432593996832432152217392771'+ '19': '0.06464929303543405613383876419120455844244789022917027481079190067446697971834722547514327827146654697'+ '20':+ number: '0.06020940656245182043572967581957492337261937064631186642499705561396077550229157404778396819126206185'+ comment: $R(e^{-2\pi\sqrt5})=\frac{\sqrt5}{1+(5^{3/4}(\varphi-1)^{5/2}-1)^{1/5}}-\varphi$,+ where $\varphi=(1+\sqrt5)/2$ and the roots are positive real roots.+ '21': '0.05617307177318694688930036962423089132354626298667960771464204218178390582565297196694090372182654392'+ '22': 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