A Pure Recurrence Scheme that enables Linear-Time and Constant-Space Calcula\ tion of the Taylor coefficients of the function with, 1, variables , x[1] 4 3 2 exp(-95 x[1] - 56 x[1] + 21 x[1] - 8 x[1]) By Shalosh B. Ekhad m[1] Theorem: Let , F(m[1]), be the coefficient of, x[1] , in the Taylor expa\ nsion (around the origing) of the multivariable function (that may also \ be viewed as a formal power series) of, x[1] 4 3 2 exp(-95 x[1] - 56 x[1] + 21 x[1] - 8 x[1]) The following PURE recurrence relations hold in each of the, 1, discrete variables, m[1] 8 F(m[1] - 1) 42 F(m[1] - 2) 168 F(m[1] - 3) 380 F(m[1] - 4) F(m[1]) = - ------------- + -------------- - --------------- - --------------- m[1] m[1] m[1] m[1] and in Maple notation F(m[1]) = -8/m[1]*F(m[1]-1)+42/m[1]*F(m[1]-2)-168/m[1]*F(m[1]-3)-380/m[1]*F(m[1 ]-4) Subject to the initial conditions F(0) = 1, F(1) = -8, F(2) = 53, F(3) = -928/3 -------------------------------------------- By the way, it took, 0.113, seconds to geneate the scheme. -------------------------------------------- Using this scheme we can compute any value very fast. For example F(1000) = 123973096123700152196608619702601381527250930928711588095639110012\ 993122719190568954390825996553595806286461496215809963727447800153487065\ 267888396644078916968866984814646017372789934941599354285962655048781981\ 449601432672314476886016051155373921619172516205959959279462742629650623\ 986561298003580415852387985592497584824080746219472211037748652055975937\ 318302049567217689699894592069801909609156368239952908226833830468278135\ 665768544331781232284249540276617296542129982828728997288152197541953903\ 447716282840651230636802771011689946378570379826438841451378240500166690\ 867529495404854605125051223062546994540563330947146207183906900379156588\ 621240712242597197632319337801670470596935760699456527541458997748564621\ 198172897996265616333729434334284714397345029438835149691252961046532547\ 719010007766358045479326193326767376099530171350877701825117365533021662\ 095429401070223484633745384385213541123394658469608329919824951743577931\ 079075070841668552792831507789117086760019602016318843661940224071181083\ 066422834306911809406410052798482671930953309095852242309841598127163482\ 618764093001518351273701686786500175853525985946934731360429433602796403\ 825145341431502282245175989711258590014072266856899442065648351186751709\ 630920777488160234982560551616906052296353246081207448908779517786711825\ 455034073347866346603992352720518102982970956451900996911085474845246761\ 286858373943501541082263778107546176749920694922716728669865285524012669\ 823663733026846082267220901703574075058073939977538088600957786872351745\ 184024070473628374260600351710919033262898991919856516183580091673351015\ 183651655174974195307254314706006954069397330713744942575643374449473749\ 503454817756050939801506815590082113610755600285003737269707209023923246\ 832369067901276020087309901539919611364073094671504148038341131540745959\ 686288313860697306280725987438129005835164325856950479373663971402468806\ 907534864113768730081469994725923781306149053657869655039690311445270847\ 846373918950946553208316893028610573765634614022749888375021991006602109\ 892962408191117647169930318421792528047080728411268209034512079394873341\ 034090361426848531119654967300677896093352472179626850291902274532829024\ 808269112919187068127606116910352543999917693667490179809549799806369763\ 488562899663544958520584072630213852256213117550763119612258142030276798\ 287393338852644580742142781789050798400564116925591559152833975525097004\ 710756099117108358727746766756676017433078459026738012517105805492257962\ / 07026001 / 12292674730184865112921275108741685942458894569513807854621\ / 834956879989888554375298113394616172092994226170364678206287398303317344\ 548260941733410396193382847386654727633570852089444707400845740871456000\ 922739081507853477360650919625155377855480727832019127085719873724522097\ 335142662851328022555976403850403311642221515906094493355369713845697670\ 019235105977920675438380166591944208698543316444489188051309266402124800\ 951708948648978180937003440268820737867218083060654797874147981933742627\ 579533987879055283387654903476911605601505603330932394841061104593740411\ 259927315629673793189546083195380149204726891000958781761117475354014457\ 752554745725902888001584048074851304338129601286475646940387326363334619\ 746282798756764967398146323902493735262848845992077309343957552899108141\ 186790949490602742972781146630743420086331311617056382004195676714649485\ 955786525734378057747330172167121490119499953334498461302227987053527276\ 928793262565477894480922903829039274212715333446091610669159061151128035\ 961328793563541195233541604397890924263650487464999719646665169973407983\ 593680967333890806619398279482125886589726030989664253943664282291055827\ 420470230581340626102664343539765522092970179642150391414490214119395810\ 712898970824676934912900697104666926204874969383459088318380326697657163\ 242417682905201615169232364953630827808284760675017349900524723859916060\ 161570681474217436901416568167220410186781701367637869182013021971045826\ 023638287259554695207567618117474263870669624755806182639836522214180374\ 545487305509475027573008138778377867438348890630996564625286541013427010\ 304880660072270327564184449810376305883796642610605687559512400706784006\ 817881159779468789490801148168318703486060839538985463126680973835483005\ 259876988504202142155715466575836835037101317457156688434391637544955278\ 666716418906848996262347540402489791214164141503725435406195262692587636\ 414680517782709984970054078432037489087151747161584364008529020819837927\ 093867074647487161546540490463309460222034534504115846987765766429457628\ 730741562725484754952606531573516406990363124650397233251345361402991619\ 536406177375532349928152185178540943361521223673862123280147940611494514\ 588066537471588119347200000000000000000000000000000000000000000000000000\ 000000000000000000000000000000000000000000000000000000000000000000000000\ 000000000000000000000000000000000000000000000000000000000000000000000000\ 0000000000000000000000000000000000000000000000000000000 This took, 0.127, seconds. --------------------------- A Pure Recurrence Scheme that enables Linear-Time and Constant-Space Calcula\ tion of the Taylor coefficients of the function with, 1, variables , x[1] 4 3 2 exp(-63 x[1] - x[1] - 57 x[1] + 86 x[1]) By Shalosh B. Ekhad m[1] Theorem: Let , F(m[1]), be the coefficient of, x[1] , in the Taylor expa\ nsion (around the origing) of the multivariable function (that may also \ be viewed as a formal power series) of, x[1] 4 3 2 exp(-63 x[1] - x[1] - 57 x[1] + 86 x[1]) The following PURE recurrence relations hold in each of the, 1, discrete variables, m[1] 86 F(m[1] - 1) 114 F(m[1] - 2) 3 F(m[1] - 3) 252 F(m[1] - 4) F(m[1]) = -------------- - --------------- - ------------- - --------------- m[1] m[1] m[1] m[1] and in Maple notation F(m[1]) = 86/m[1]*F(m[1]-1)-114/m[1]*F(m[1]-2)-3/m[1]*F(m[1]-3)-252/m[1]*F(m[1] -4) Subject to the initial conditions F(0) = 1, F(1) = 86, F(2) = 3641, F(3) = 303319/3 -------------------------------------------- By the way, it took, 0.042, seconds to geneate the scheme. -------------------------------------------- Using this scheme we can compute any value very fast. For example F(1000) = 890009451435971423889009064328239548949802596311362520432404428798\ 733421509981365448094061673783494031050442900430769597825217073245172304\ 290055696298322899620819061486797390253265064715673881170433950156506122\ 923942000253246102078920577674201856429633927299777316251786760849565604\ 908552938523711083571575182349083834925426887064482691971443123793503661\ 039195398129176505736806379882720110678984855053292928329852097383050357\ 669006955049689724096517401127045369605315296331793870294143257730535865\ 879117410730956279276473954400250237165200595211452907504939777008924638\ 062639313159469703712451064261287212473332392652296693483429140786962223\ 868559176502589855121978061073215892547815174235290904168088569348901288\ 291709708236175200436294730744085942423008070016053187018207357117571208\ 861688553421090197876479508238461711932544614438282123354153858210247665\ 776716917851330882711672398206082306976522314268962331026701341227960589\ 352494598114165137080679829055241830248623417126994934391652613669909637\ 623018658012625107936324042399775917634419857783275432503181834762702926\ 214511170749634242962835029220729683416248501586057620326722869238346324\ 353879104481452620003502636488415294782193583101299794806907266430172845\ 270773247419917151044482899125349686421022397645555105672005248760143155\ 375880404985840893163865301985225654663220010083088211458251258056378547\ 220257102912273322197377595494681100138904005045679834096215640991601307\ 699609068078641975446485712199127793296713293162719643242279773424647087\ 503867477533678879556075372119915137514442173496374023703296916177203211\ 886603949615790175438159244682139654937246307632332208675704227889701574\ 212994589218703589061303016037699698497668456278126600135550737251561371\ 502410719641707226866744321702603332839660239845343321600567252905097310\ 672718135358220543691175472348270571405605551572761008950797040250563331\ 544291088540329117296704660578364242730596117093718577388873149720799459\ 028602523653099478812060997274446953270934320490900445904042101871662484\ 508810779214688905585827207649636831965362001849949776576466498447549723\ 741707809455611064888568922990154772423525637349617060564781808054748518\ 056099706455931246135525410764317824760548478131292967590551741692539519\ 209421313253780094874298070239561950530532279235474077691223136997727180\ 931114256174953823397793998857143830300304995116684580171634706560944511\ / 68655583544071224588101 / 30762733501098742482570261909806741141697566\ / 427013832838524656058624484055385079282157677009284923225738445196567237\ 874041661127687472497225023924650139179699717202640117102617030839323156\ 345402498838081448674973368944007246581863355503224278609216676315421042\ 394554615887035842905981938334924438138356622733684448063036616189066036\ 359085664611940184889844428476587105803968948502591378342410856324710039\ 903163967234233002212030374383186864901799034511670118244334912325189707\ 504924285780404284966573983635418356129391008073751807375553150086117050\ 782035380345109471034982795627553391056810720216321660506569179671292780\ 780792482256704189627444290975209339986648932111132041406380821424295192\ 649290112704504584997180610614443498977312817758219997313417042497059277\ 314832595115909498785951041935032630731069217549935174369899814311561260\ 133811276278685449319770889561663168904366759975093375810227023144008162\ 895755094746450186236697417361316776790580451469751108919022233042563990\ 772469983696940806214118158998509471953814794090735883887797398163211886\ 641849874472040219037551366724619090281350243221234633578779122010808522\ 300689441995519315446521451855512221379266984358342146377439850596644092\ 548715432794435752713813096918343188367227288959585951592217941289543667\ 537335282216221979130054965033726359520992814842120735903654201514917391\ 429823280785705726747869290624495786402003016885079054270455062310722440\ 842486066835324844017638807371423441688018835122094527004042969915224391\ 360197544576094307071400818721517378983643675709466181248683080723033579\ 216586641869522745784423019999732173701442128028882670288918280892119178\ 476910965120425255296865653664752800248326418015139679494079209472443410\ 985731992341754183470195014792341452718611383186787932409457867835328206\ 720378531988703490706944770415008289878336938124881619019127777465046754\ 885320628218243865616944197369193529824914881042638187956108735102879140\ 729767278942707759398599066592334326234022032758040369897793479541798165\ 600235849805996423004514514396305177471750524353136718107950229148399407\ 166805617969272555658662459512965680870722463116124553654180754262840288\ 717702066570526720000000000000000000000000000000000000000000000000000000\ 000000000000000000000000000000000000000000000000000000000000000000000000\ 000000000000000000000000000000000000000000000000000000000000000000000000\ 00000000000000000000000000000000000000000000000000 This took, 0.149, seconds. --------------------------- A Pure Recurrence Scheme that enables Linear-Time and Constant-Space Calcula\ tion of the Taylor coefficients of the function with, 1, variables , x[1] 4 3 2 exp(-84 x[1] - 5 x[1] - 74 x[1] + 96 x[1]) By Shalosh B. Ekhad m[1] Theorem: Let , F(m[1]), be the coefficient of, x[1] , in the Taylor expa\ nsion (around the origing) of the multivariable function (that may also \ be viewed as a formal power series) of, x[1] 4 3 2 exp(-84 x[1] - 5 x[1] - 74 x[1] + 96 x[1]) The following PURE recurrence relations hold in each of the, 1, discrete variables, m[1] 96 F(m[1] - 1) 148 F(m[1] - 2) 15 F(m[1] - 3) 336 F(m[1] - 4) F(m[1]) = -------------- - --------------- - -------------- - --------------- m[1] m[1] m[1] m[1] and in Maple notation F(m[1]) = 96/m[1]*F(m[1]-1)-148/m[1]*F(m[1]-2)-15/m[1]*F(m[1]-3)-336/m[1]*F(m[1 ]-4) Subject to the initial conditions F(0) = 1, F(1) = 96, F(2) = 4534, F(3) = 140347 -------------------------------------------- By the way, it took, 0.039, seconds to geneate the scheme. -------------------------------------------- Using this scheme we can compute any value very fast. For example F(1000) = -83719449741198877932493998388450695647319146531957116447709450355\ 218179869789491886095206684800653276874060353574442534347149322732346553\ 389102421329526153111090966870549594741620348896976396541944122447624844\ 566008680378175873981338257792849718381143106760512036025464054512715396\ 052915168192637571998903436085627470041072933285656844540821514336821300\ 257192733196387543383445229755839674587496347142505906128515987131426626\ 737534602997018713953488434705064379375004631900072279929227030664502231\ 211982504573513187873045447442586302079659445930560789974847219824311583\ 684373510265072618284056068650217664594459766291800218376950168840405391\ 083229682041168233541937767670288381779883374197640843339096219704000720\ 384165462790115235543898200664310433226638573452372035134896361543186929\ 131569208858938353691496216120425934161750710013095250477343289131615582\ 419736069509733572160755310720623486463352047271931881340421073764895345\ 158060868627948821423255249148608226944645553154329012499059919716639682\ 753580367134561840948707895104120377962798228588098939287409524215042160\ 423323462861372253648208710596307259308280225628082318493700761923847255\ 497159029020805250762758682173318750766825805210953286315725768947587419\ 653074361011997937237697207054276420068854955605013190544174844134850874\ 663544373915187513492244490291023896098520471161144645740241073994753033\ 427592415829697978038555518210852698300092618949063574234935236912140346\ 732035799640168463254578151105315222098727394425369951519875488492235156\ 074449949372261186532131334529023659150644384531467983607232961359081300\ 416059710972709846886310693642480386308873536647567805164684238119331324\ 356723203779684022733881178564940418711783590932178148071819682347827453\ 039833627667039900271048068778669289620027504808936422561399319398891831\ 878311281649929754082384721569337544507726080479174980926289292949533801\ 188041482656392933352348444508725326823994430750003124581565271666987537\ 910272100581137802438917810682918232196055166191108736717711814675638825\ 026462262200014657607945904546661761048739890497823515713538936786584633\ 462821378247847465441931794130201542950364222524400971743931174976181421\ 139894105404886100295786095042222983480069533078394426211515177292204226\ / 98583012480478648558778447682904645571484358262077 / 21442752520599246\ / 507819255943002250093972378694610107964858317270001022616208266884749469\ 284417582177514234817804775292393137101687225285576988752320207215296154\ 839640794634893386876689395817957853406197827333061923065566244825032385\ 524031401903060182343518009063753031811695914411527647714696426216106881\ 722410952683174524292800380994418652056319896438472147671251259987485191\ 036909802113122646183647158276388798312503211165191543596429791883797579\ 861072137027272836083860268676505088817219517022936840529500807514844665\ 018487905274993445334340945274521372370337373847713423697367309610202275\ 587096343895604131013077932584262544763185037005040603695951931948419466\ 370860225805710631640282479542839304077626450080136161515585209057254241\ 386785805814888979384282781006874616811195758031770224696939376891698182\ 399119410623390485951726730438922952374319533378705388688133883265117747\ 488940558747697470843252857273689998618199778176243104933838786592154318\ 590410675108569083515236378715816482149600611120222927180375653295141620\ 791830101365873654693960368482551969290312200709165605899074080714350009\ 625992382697136910721399845836831698173277799421704068412881010079788685\ 799658444118811952885792072597726514781951669879926882777011975126313236\ 844277926777576921690740816213082181461361462198140559883169015966173217\ 113416856134549122629478596852216801879834460031860286520883719021748358\ 720974822395120306629323197818311872804822871719026425615618832080942841\ 580211486833799283436913321354653298562536692315185771608120851309350655\ 478144001868977198162688622540137665425911543847497457541755401480508876\ 280927002178510268228890829587092904902131041246020374392554546786931205\ 223025921949860706792549416022373556281286050926478987015529794977057184\ 440711546433610587313936791545426687404074723505303086766078826869522768\ 547467102222712761719881928731711371466284986247258337098117885128951481\ 428086839722629612058715531265683614752636745381766977055424840093056369\ 554258458101831663344711101946265600000000000000000000000000000000000000\ 000000000000000000000000000000000000000000000000000000000000000000000000\ 000000000000000000000000000000000000000000000000000000000000000000000000\ 0000000000000000000000000000000000000000000000000000000000000000000 This took, 0.121, seconds. --------------------------- A Pure Recurrence Scheme that enables Linear-Time and Constant-Space Calcula\ tion of the Taylor coefficients of the function with, 1, variables , x[1] 4 3 2 exp(79 x[1] - 83 x[1] + 61 x[1] - 11 x[1]) By Shalosh B. Ekhad m[1] Theorem: Let , F(m[1]), be the coefficient of, x[1] , in the Taylor expa\ nsion (around the origing) of the multivariable function (that may also \ be viewed as a formal power series) of, x[1] 4 3 2 exp(79 x[1] - 83 x[1] + 61 x[1] - 11 x[1]) The following PURE recurrence relations hold in each of the, 1, discrete variables, m[1] F(m[1]) = 11 F(m[1] - 1) 122 F(m[1] - 2) 249 F(m[1] - 3) 316 F(m[1] - 4) - -------------- + --------------- - --------------- + --------------- m[1] m[1] m[1] m[1] and in Maple notation F(m[1]) = -11/m[1]*F(m[1]-1)+122/m[1]*F(m[1]-2)-249/m[1]*F(m[1]-3)+316/m[1]*F(m [1]-4) Subject to the initial conditions F(0) = 1, F(1) = -11, F(2) = 243/2, F(3) = -5855/6 -------------------------------------------- By the way, it took, 0.040, seconds to geneate the scheme. -------------------------------------------- Using this scheme we can compute any value very fast. For example F(1000) = 426450175569362852192735078850554081466255681799505826488017007993\ 532700259881387213823570832251445008648252963854016865665557526247739476\ 637682178261394106660096462510760188557651939239058400976059295672529468\ 980921609021753495091455492626138489578702167082969262884611443920805479\ 238888401899325526554194467700305952207539657129383947965124091135867364\ 075987541097203135059409107002124817845936330111768663471158661386117611\ 363518196112302209254056244782591438639362101151377717398159924395481734\ 762361540267656652734172974520246435011997689072962182575888127744704684\ 058726298605238437749544719414476162194216530560770072792749294301471810\ 046711264019499692323214458488014094547787553405992177362640513880361758\ 071576103569898467876185188948420442546160799313955110479951165509824952\ 346473611112842794354253864663047272102106459604207718204098379029559043\ 049096749447462518060254755294692464105527156494252420793394824877856468\ 445602520322370346249048204631752359149773241628554484012518666220033694\ 037922993275326139773460775207017640446730167068372283383723493164893501\ 688774074967528212133517883622578291078325886490918573366100301014384416\ 920384576471565274604459752789514359468262296998867986567628290474108527\ 797810464592337949798272538403980596909214993251003057928206360268666056\ 734896958953686160186742509510616451684490761698332196295217816052651889\ 951216074731531732881403912442125268037139966353917238428853925952317264\ 103549615585718304207192604333578878876555769593549633400681947555246717\ 010150929884451264558104034434391652182562965889169960127530371127817435\ 945351993400179014977620908142790797763725567336395630759694877750648944\ 304138264606741521884838608081385294310120450095759612122291162217215267\ 050723503796741489074618452882788407724434708692571752453878654747006580\ 868145007392642392227012321812404474595395182513670175842807448916025047\ 646084131319188274324866472981546406130963304690233060034621184156095069\ 656742332729978762072770694199619696859997503840205078165614920089687112\ 314994867525936087313171792828314723701121726114024479178208039068120717\ 526115858533969592050450338302834902642605579530128416695476548170486022\ 572278528437862474595454099649805595223031748090482375049968152129681028\ 213944510622333454012397168393798655603164104523886955336304664485689018\ 355499779596715574070551064890374219591230230224357368217039745510631549\ 509077129438610193750568310877679576237447780134148727427408734581449506\ 316462768678716865431577668176253955651225724142455650263920258026391593\ 869403353641747280418832726959230152789128475040219174417778208054836668\ / 60035450584535282399 / 64015826147757769578189697873741539512707718063\ / 352493747393868123090816691771979992978846378469668822367728734284015174\ 804203553678009451665409219987944161454021866725213282043478356528006168\ 687173729705827286290809492860215859506311726989242143574660728654767644\ 596905755024183518095871434156909568684171276950036248248113249075482509\ 252812395320048556169792115339225086580987649308288711165746090090460508\ 141200423641090494030934907245400272125817460427133816432498330362580526\ 339999049187453494792105065077422813886721935638080807992578909450011408\ 439361296585905707882864464741231055760248571473074775807900165801828950\ 028523681494223708636023138789656832959720623823101403743210358620537879\ 918662041923478625119749262743678728352530046790085998424555467277428972\ 369543788559490304704932406441558523205018598052785890289025450707453091\ 992034782998756244854744135268876208753486827061251206711120171096913005\ 399167615411583604585326499937292703817955564051794202182867524621531396\ 016744819943049303936128579749788838502262000111643227672071887774777757\ 960660567710429852864041361809791933969468294159826372745414399852503666\ 054203840462921099341416496258381402227666348117844452737345427782062415\ 403780076253554552460836179528126132738122242621145995182269730818060583\ 774807836610709749389645616462827033536239680349254164380686579207414836\ 276549055440923746544156552406265574449800922035206570280220096388993906\ 938812165354626324406220465631101504854207070626424436993693521396048885\ 300312799911182369384578713020311094072929033590142519819930608112952006\ 049865635222878133436451596328454229384064152148423158299219633946320781\ 060973616419975242001495333712951409719101630997716350778993070411299289\ 676707810213114165964395227049621302003134542231529523770310924378981227\ 882044394846540053390044580873415310085742884491383557441260286618140482\ 303631979902977466792260736560797868517300613138882212772730596071585425\ 568012061609718498715921532376227265922750684007927262812724469498916832\ 724909659212605497417553157336004334779463120994144258475863373975173780\ 672537737467278952825407359729312890738245506061702412820401901013248463\ 707170980360671559180999242988829142848954713377557422491598953934112087\ 093924567451455045686102751925085531387097950283233021132800000000000000\ 000000000000000000000000000000000000000000000000000000000000000000000000\ 000000000000000000000000000000000000000000000000000000000000000000000000\ 000000000000000000000000000000000000000000000000000000000000000000000000\ 0000000000000000000 This took, 0.151, seconds. --------------------------- A Pure Recurrence Scheme that enables Linear-Time and Constant-Space Calcula\ tion of the Taylor coefficients of the function with, 1, variables , x[1] 4 3 2 exp(-76 x[1] - 18 x[1] + 70 x[1] - 45 x[1]) By Shalosh B. Ekhad m[1] Theorem: Let , F(m[1]), be the coefficient of, x[1] , in the Taylor expa\ nsion (around the origing) of the multivariable function (that may also \ be viewed as a formal power series) of, x[1] 4 3 2 exp(-76 x[1] - 18 x[1] + 70 x[1] - 45 x[1]) The following PURE recurrence relations hold in each of the, 1, discrete variables, m[1] 45 F(m[1] - 1) 140 F(m[1] - 2) 54 F(m[1] - 3) 304 F(m[1] - 4) F(m[1]) = - -------------- + --------------- - -------------- - --------------- m[1] m[1] m[1] m[1] and in Maple notation F(m[1]) = -45/m[1]*F(m[1]-1)+140/m[1]*F(m[1]-2)-54/m[1]*F(m[1]-3)-304/m[1]*F(m[ 1]-4) Subject to the initial conditions F(0) = 1, F(1) = -45, F(2) = 2165/2, F(3) = -36711/2 -------------------------------------------- By the way, it took, 0.040, seconds to geneate the scheme. -------------------------------------------- Using this scheme we can compute any value very fast. For example F(1000) = -10619365638602732749355421312774404719017458004081381358439579438\ 987995163896831123660640598500491306496653723595624004877047893511888406\ 184635820743457579677777259405798970143700068175014795981860448758398039\ 266746717274856645160289637685856698743229711311547285909319096612956523\ 727023888603842725369570170948633684200922951096699452583383046392687098\ 200249483636276808376845680092612481095346055110760206329118559377859488\ 877977393662789365190763840079223528270524379572470083166117883099473536\ 826582773825486541369115123281215421991151267525430126160638732104529050\ 834469754908212454159685942764907556407644630656841888985776334580905792\ 508880321449271383742441913273310256154496994973133021785447333719106153\ 741942099562510971696773678802379020199121050658101109679543026679413106\ 347309355961765458011160357321464473147851344182311974379478262474712181\ 168330390571483610698218032329165326305117201542543922574674245029623602\ 784515114493947588572233328111406762054503962791517221082433753412205114\ 541580810160657557117861908136720462798378923066746933244113600175236957\ 703350336279618226523954874520098714282316647940631556567571098015346575\ 887239894163226940440871332842195695016662340102900974618525791086832893\ 710470796646758842983511912641556058368290610155503431852502070416602136\ 890642739586479243049133640612037404155514451249051314248828838652165900\ 904689786800622919442731352421188749857851306902738844161755723090796535\ 435001946840983807343810639034080861098391365109653346181714775656310312\ 011684404522509812764129741751039621779921994560071031572105239737101940\ 875954650142853703246018357440893661712185118859143121463255545028168503\ 912544067991079687058368774192117167091989944632369074979770706955543174\ 992097406863048037250508670607374892808248814846008754569973503854192112\ 459625625632689067200600438135613186853672422793764604080439909358137476\ 486267117731457114230185775110151099643844607121460127854466685714306036\ 861159579328353547389995120697945507816364027711637983278693097388899675\ 326793368237585330048990271727184892874763396314837312885137504912019202\ 988361029716730903760411724158207310991297047893527030802988588456797517\ 907520306328602296821773807062949519113413362881870700017780228376811153\ 928607226658002897417628933298742097710304955489552372130119797555464870\ / 21007824014958183 / 80984857047720815934669782956703867380226905754129\ / 236670784115941595795489222722580209607413387369448248089001366433540829\ 213325815751221133537281003235277104650669716822372529425835623593144039\ 275527205959353843614908766084691630688476210170083362459565202379730408\ 591199007412194114125876185556252271019478871744665442641624812080895309\ 654078612150837094186779037949274443342801445202835332228464545989164073\ 453977445410328931743086155647504323910712714547477086148495014868920685\ 516098053324486254473150169905819593991293194712442338971405801510589510\ 090172833831900367516960196238939635447819034740374948436277376075981227\ 623729932811436311328626295919905590256366236416514042935576389099246349\ 941349729991044965306383566007035659873099163909732505590468517610064407\ 013632824028376474517363820710565171660548479465285217697269448524045434\ 060157464253799532015171471075380547214284258862662779324210159091406794\ 771546790165147574060784146610154179329987464715312020164992983367340361\ 252960075162384379237793718600985857043086187329645419837446115260429599\ 604750632690065508029766626677622344149096587747940305033430367971158920\ 020006854644568535129331293564327899492366877218705450297743396280144864\ 264565167436283058279440445965945325886978616988700481769622459997747032\ 078272663514053087845212188485027914146819534788098835161480514849060974\ 694093110384334219065578490084921782711824473365399561007555716689166048\ 755313299798162451182409880103146767658565868558574605048211957451652749\ 177946300274993935743971284702798191525552205259867133006593961800083002\ 339660664056313544336341226610143242310992370548049294147934071968569868\ 546191069519913993245213133150090385530896382573544633109240695308538830\ 325796517886279073577650840886879233242640489451508847721365275177740550\ 451989185593827218070201233159012744099508496519353675130180616009387874\ 377039143195541024159146430041202500644513351583828571557669890087363273\ 917022532165359339809113876891989736462312376317012285384335598307843747\ 418304177439531059733325944682929159335287356200134662393700964669413732\ 488198439516400494567802449857270361467027801194600671815800418200545661\ 282921439811909503004539133475440137393940576203077542607339690822352574\ 454186500991851354553575695974400000000000000000000000000000000000000000\ 0000000000000000000000000000000000000000 This took, 0.115, seconds.