NB: Recursion

Concepts - recursion - recursive function - stack - stack overflow

Introduction

A recursive function is a function that calls itself.

This is weird, since it does not seem possible. How can a definition refer to itself?

In philosophy, this is expressed in the Barber’s Paradox:

The barber is the one who shaves all those, and those only, who do not shave themselves. Does the barber shave himself?

Formally, it is a type of self-reference, like This sentence is false.

A Cute Definition

recursion - the art of defining something (at least partly) in terms of itself, which is a naughty no-no in dictionaries but often works out okay in computer programs if you’re careful not to recurse forever (which is like an infinite loop with more spectacular failure modes).

Source: PerlDoc

A Formal Definition

In mathematics and computer science, a class of objects or methods exhibits recursive behavior when it can be defined by two properties:

A simple base case (or cases): a terminating scenario that does not use recursion to produce an answer.

A recursive step: a set of rules that reduces all successive cases toward the base case.

As Seen in Nature

Recursion occurs naturally when a process applies a rule to itself successively.

We see this in fractals.

Infinite Loops and Stack Overflows

Every recursive function must have a base condition that stops the recursion or else the function calls itself infinitely.

The Python interpreter limits the depths of recursion to help avoid infinite recursions, resulting in stack overflows.

The call stack is where information is stored relating to the active subroutines in a program.

The call stack has a limited amount of available memory. When excessive memory consumption occurs on the call stack, it results in a stack overflow error.

A Note of Caution

So, Recursion is cool, but is expensive and complicated.

Recursive functions can usually be implemented by traditional loops.

Example: Computing Factorials

Source

The factorial of a number \(n\) is the product of all the integers from \(1\) to \(n\).

For example, the factorial of \(5\) (denoted as \(5!\)) is \(1\times2\times3\times4\times5 = 120\).

Let’s implement this in code using a recursive function.

Recursive Function

n = 5

##| tags: []
def factorial(x):
    "Finds the factorial of an integer using recursion"
    if x == 1: # Base condition
        return 1
    else:
        return x * factorial(x-1)

##| tags: []
%time factorial(n)

CPU times: user 3 µs, sys: 1 µs, total: 4 µs
Wall time: 7.87 µs

As a while loop

def factorial_while(x):
    "Finds the factorial of an integer using a while loop"
    f = x
    while x > 1:
        x -= 1
        f *= x
    return f

%time factorial_while(n)

CPU times: user 3 µs, sys: 1 µs, total: 4 µs
Wall time: 6.44 µs

As a for loop

def factorial_for(x):
    "Finds the factorial of an integer using a for loop"
    f = x
    for i in range(1, x):
        x -= 1
        f *= x
    return f

%time factorial_for(n)

CPU times: user 4 µs, sys: 0 ns, total: 4 µs
Wall time: 7.15 µs

Compare functions as \(n\) increases

Increase n to 50

n = 50
%time factorial(n)

CPU times: user 30 µs, sys: 0 ns, total: 30 µs
Wall time: 33.4 µs

30414093201713378043612608166064768844377641568960512000000000000

%time factorial_while(n)
%time factorial_for(n)

CPU times: user 7 µs, sys: 1 µs, total: 8 µs
Wall time: 10.7 µs
CPU times: user 7 µs, sys: 0 ns, total: 7 µs
Wall time: 9.06 µs

30414093201713378043612608166064768844377641568960512000000000000

Increase n to 500

n = 500

%time factorial(n)

CPU times: user 494 µs, sys: 0 ns, total: 494 µs
Wall time: 499 µs

1220136825991110068701238785423046926253574342803192842192413588385845373153881997605496447502203281863013616477148203584163378722078177200480785205159329285477907571939330603772960859086270429174547882424912726344305670173270769461062802310452644218878789465754777149863494367781037644274033827365397471386477878495438489595537537990423241061271326984327745715546309977202781014561081188373709531016356324432987029563896628911658974769572087926928871281780070265174507768410719624390394322536422605234945850129918571501248706961568141625359056693423813008856249246891564126775654481886506593847951775360894005745238940335798476363944905313062323749066445048824665075946735862074637925184200459369692981022263971952597190945217823331756934581508552332820762820023402626907898342451712006207714640979456116127629145951237229913340169552363850942885592018727433795173014586357570828355780158735432768888680120399882384702151467605445407663535984174430480128938313896881639487469658817504506926365338175055478128640000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000

%time factorial_while(n)

CPU times: user 85 µs, sys: 5 µs, total: 90 µs
Wall time: 93 µs

1220136825991110068701238785423046926253574342803192842192413588385845373153881997605496447502203281863013616477148203584163378722078177200480785205159329285477907571939330603772960859086270429174547882424912726344305670173270769461062802310452644218878789465754777149863494367781037644274033827365397471386477878495438489595537537990423241061271326984327745715546309977202781014561081188373709531016356324432987029563896628911658974769572087926928871281780070265174507768410719624390394322536422605234945850129918571501248706961568141625359056693423813008856249246891564126775654481886506593847951775360894005745238940335798476363944905313062323749066445048824665075946735862074637925184200459369692981022263971952597190945217823331756934581508552332820762820023402626907898342451712006207714640979456116127629145951237229913340169552363850942885592018727433795173014586357570828355780158735432768888680120399882384702151467605445407663535984174430480128938313896881639487469658817504506926365338175055478128640000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000

%time factorial_for(n)

CPU times: user 88 µs, sys: 0 ns, total: 88 µs
Wall time: 90.8 µs

1220136825991110068701238785423046926253574342803192842192413588385845373153881997605496447502203281863013616477148203584163378722078177200480785205159329285477907571939330603772960859086270429174547882424912726344305670173270769461062802310452644218878789465754777149863494367781037644274033827365397471386477878495438489595537537990423241061271326984327745715546309977202781014561081188373709531016356324432987029563896628911658974769572087926928871281780070265174507768410719624390394322536422605234945850129918571501248706961568141625359056693423813008856249246891564126775654481886506593847951775360894005745238940335798476363944905313062323749066445048824665075946735862074637925184200459369692981022263971952597190945217823331756934581508552332820762820023402626907898342451712006207714640979456116127629145951237229913340169552363850942885592018727433795173014586357570828355780158735432768888680120399882384702151467605445407663535984174430480128938313896881639487469658817504506926365338175055478128640000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000

Increase n to 5000

n = 5000
%time factorial(n)

RecursionError: maximum recursion depth exceeded in comparison

factorial_while(n)

422857792660554352220106420023358440539078667462664674884978240218135805270810820069089904787170638753708474665730068544587848606668381273633721089377278763127939036305846216064390447898698223987192970889621161265296832177550039924219683703146907264472878789790404754884162215226671928410969236910449565971736352948400223840381120644820230857671104502306174894755428309761781724040805324809927809328784055486199364548291211876258248802189173977900050213212598043639244626460770511358846595108675470585833924655225589035474435988347383178988034633008458631510209091509935653820010933047965742556741930917055172805200236075085991197635228755907902043369743123506916831211924495971556267407521462198986233088625998302859864857578749445963115286970886710046268423648178989905454690861391613218344174148807186234448114831209490361196546872767755617886828720269104814092456410341835975604276458161513178575901661071782544156980883359372729995603371371200471049437656291142488605335299499642300699972204918120100819059439140675053265004775533850899097945101551091486907004407119572336026243368132330218709287699196806656569752790422258267841561083376425781032629202687211070274681394351128601502326190649959171897364176378436491219709109840944514895358959103804176941956657834822071749105512752639148381172052604826965162642710094919393332661030104360530459117014557209584714353721948246686793467375904872268133410207860903657108806376616249749507413107077401682180585945526445171409277469230062697511346044174567946735828782261629584248675157379172942724178783105429858245117575511884506574424827574660800238588378492396247368761507015767725898321128632295537044902516387925127590841791744640466913531047347984464996154595542013996317357476301740036796192919942190762895445656261767041799538161133387312823511534152581309087915883638351664797225912944270653557142511737323807232632958121797916679692329687096923901003255574789055099807487061047230646195984955239657612208673866514171699307557691897902675157342075864796345338446835085965490727326321910504064289713096224505162064669468098869917122127404504020684923266241760132910227866687270305284709452526825496617772499645206699836925910690894082637401043498371591126455822280606361394115344316771769934353664284928294436414769615881993661388255577487709937004594753907845149034434521174560594039916268444697661821387470705325559577933196460996662145377564935474169708562389214773222865507182490430016186142192760452307670621142961767274704123616107220009743758647492753665149532164780849075146330071016691313420662882562618283865836983632108760710427516073348347788414796732427080410860761841281888307115098982135338406610652147087046874760995427473673509451553599769040367353385551052571682650317682405743993414862392331981432579182193321898940450865013610998098383993110996355981328001049731588596312131853801205046787642910669365600437305633431984879048998524701293300789344532868156679762880495532846386020133480265279836946393384995675049993707814746561543438930431384237878981847802886009971088695632988347711863122382785963653115132377931373647397429369411499028751972227999545182615488298951151926682112451355318472209990435355949887299922035062039816011086376236539782172380237846650673624510635034423187315338308212043804710999419227821039747552717416043890169723961305549371844836119803565896062025009093664399360172007383613354405094329072476518909502507724675841989412224659392163116352038147362479528539732089309533421910635702805576629720156556510767780805933453631121829561792887673002802450932122778852968418208261778476955644980385691275787372678040959158711733971103165232678060798127609246173504120182666874262805385275843979167609007743380748420751185119102921960339376280986753665085212869255321536787932521882574101866137054328973735862725370178558806639851350386944039604928258820180419178073649693885802597758398892014389747165465973510852605706234402069637065660129535734043582961473427275805630839510667375349259659518575646939723218275780003250593895303820539697558870511543073920827422440516299708739599768461206246629098112368012579891284802505094028916959765079395437191311379314427405135599630375642214527294341797246187597964074239147838993541565834716156858499036773056611353833367087548900413091981676330749041510337597307246885839246941715548295730750618505881581959528992660225626903439573313450666972952115230668696227920947779974336574472673471408928071411283888082693377378077293104110767513639476200610858040596019639058015761002337463869352228385801434957178125581445862930042479404065736859862007914604590255413929950088044710384758990326548097338166940500085452723713571394902463820308668541802838317527668064278489561005755859991718966786449154063570014497194249878920859731254275567514575206399118150736397483102490793841725653421894276769116598153430084637087769510295415136551734675054015239706042571746001089968440498845985477977905031632568489156557231006499726498721480800181770357701502983008879487243887718884416833034708723239505377642232944095773219137582371673924704216723002256883135779230394688900662466182532658490724406767024939579697217467485562998183149665611743997680482094166257463879660305171274925119226367615337524381656217330771650129520988754856467131862602387619964334867961514408328902061082833180891221325853682856469916007952105166960451695430614212305743006877217407155473217957577017595967640563812729153867513698712395570542350999228605975469962186195531354132139126436676900465429996811680550737866770665988027062972502001882845886145344368771455361304414465613369092862748276981946836480550952968681758714859972973082332924094777085275279923304892719633314751563311192746150389219290616780607901383451137066300684376267199885515143681266137319912103235469786756421210624899005553564022924345831264231038363416781719908354140411717740185950606674198348143345444247191436828225654380047860390575922417071802670646875454211626958746795398540784464654140381751149965273621123540880166990280149033225139460832668170930713868826549977374286127789417784752681328371818759103642140881783220739808059714203285309721443041845459183002833408705783138284973283761286182927136745161897366207237396132790944984014154408304074405393067540767126182547597130843470311389815695365971788564022750674237400323621850094765267521941901241387478279883426470873616812485384444012772521050072279315853096279121131160167772077952572613800240684421885453537121341902236379684012385255288607189967725694227433323948595075570839061877450159652184414998155476107548008054192318436948191732631430603548399790783307267636729090980772827355854348032260067472537097785464567761181807367424367391769863758072145859791485033700592994963793369100283444558089838054017635403737133019311293080958287612107380374800660269767842888358265737486556785868822015143046249655995760379768685318192365806469199584071845493606922169776137542662239658644989770921478134709127917460871630220821981434654245065731262683089579031012893360788644107230184805400373136014216229159146992019884148290014414312800903102107833305090238435726779416177246873411503598700003151092815700331081727415624680432977205070450456683898626301702989301145364477416856732512330376477881749036052572605520684370616116539755132541369303867783267208227323664249206432363089268768826650939691861683271739757479552993242406186992420363781929485368098035256331092448215269276219116259145886393677034653480367887126133367116968226450914997055448521259751870084720025674658752403932061045903070039438252019383102480929019684602472171298321628237994627125366359971898374425099120673688383738299653892030662843074547559074235345274029211606091346327684749522046010409575607348155101677203187580089224494752922031093841661588823584993931745149914395557357641584185479831702428523965451087525425464777294595230360946416541997797947136806344915998772409176443137371178542210740572121166868692153240490080384205921192622875440898261478908123698956367080804687628524499897440855677969456909042340530355943524640751677873953113928698614347227572144946891896093294375476741234907792754338349412323060078767610089949156126934038921148370217193387617823370358925817112869563450001367619897145400986643461922197676975930010555225198913002123021780831934330880446592954552165911855939202579781122952065357362914478404946474565003115498072056580360667380889572746464375428055819322299305089287806874537401327100274428317925355003451536693172112088227603942809788645727306979971285649576934354004030728440581746648376658498039958964243370183454151720285337810904113124462432903353964296651109482836884580127588701293156099225044518125460113274986014470437757313881001319276124676116614833528935557503106018449788994378274613854651708241613167681463911870000812845144341406739985430072772303758111613511094355614896323929750846383152930263582535361784837558519666949972251935515953807207838615142130284450051795239760968433198292598921623223582396390262548856855875458198371559008447860086745945709118128793228222051767509371866110013193625845223493949829511199280837860523506412769337548130609594264463425077601147334209139128541628183172262143783062962408149391997187528106367348876678481602342743230027158192404187686545826519361990687336892886715133840245486110982482004482721799496658712257174429044916781194824165631560303473833317665121218052780795958220298330611945164019413315550379662980215357680731124530585915969709973988055743550083279071844959752353594644354789680372126344509423070253995102864458237454677761013556916212309752286152053213998745673034127676503369636682306665552051562491132528926155863868503100849180920507680658265915276163719928694258350604859732273949286080260640627521341007801815105623787926212039424781833439433877206395801115809084190794320195178235740190546595990289617711776195270354051193727229722248442080440098750369411277686593022133010625031862085145076421052980508837197986052557750303949606158442838846866137510968441567309838079394349570013029265177957120625555851951313574029897589283475525334409858911400694449308432874005015554332587793895080241128538758725945136400838324944471346436826148195406004114845870234072926697740631325878634790667698266181501256117692275715291249164821702372884416357600996851100939411444677628186007072278522314941048564396255796808221289935799262208553889221164765220850367706476491496133789353761537391569177822237744837614120253342622508007300513473422771427333106345971803240244226950458090539326689103619381998838844036231795282435495362489670734155948067688515321073064476077859628627852283657244564306449096277517172656954238392941958409527253281659572534531428389629894005886539486824117113929627356938973482935854650278689437014798383826002058208853517073216288725214522205265969614962147884840129004507737252424605074339660818182960296019196314124998538422017695110361380561701016357743542531148669369994130940908368220071936435111978592782493491477052187226546109199597269439152400467901173602521030051886080337084840114810246351288263986170081804888380750203521448348740849154718714478857809574515499505005070789428842888410027877777455981132319940624176532148686316581736774410084063436959989519288310869124517866342559353458242589411390516469440377562665821577845936829909679754548350510473633770839151033854639602753486401635204633884342346714935641429160856846724874244782055113759168236472297793612971080302530934478115527737540458968990354808058309381267323593563098546564376209385371052808344607189076003388781618019853273759498566916704703448438363503416368325266403224174519476678140428319327482851882140344319384445475456765253419659194332132585432270070759038565239668227171300009189122050845185261514627937717597528852978637931711212529529443323757910072909001703558763798612480281463093944391916950129336315045285163539312868586427437296109446101235604877439863299611899755965996608749049271167685268675335991297583209089553296409523640116060078495005377892783750147344122123777907727134146647404489837589487675423294546899354220341669961366698976529978580795899055864050388507083137333076839766882463680992355219727241831735127646189112380485883115569477888101759708977682149644340317924443085170303692214137621194388641989508360339306459037361842937028710758321966607546113761076362543928614316242890754021082233620012309384737312220374269033838579928678572939434168287053763374091938184632261131740934278117918891642447513543478446040549455379834556163353815868441692054518698919434175386663900335756765603264363767906721626620330878425545157208117246381251512669846685887209013144861632560461019513371814585249988176629925142145014710206193190373671380347663431029705222414785030188275106347446241258707937339085095757724316735066885094208761536164440443755860160625837091300574162065273670941888667964570550744724714137001968165215954380698515999483361357521322106131884771926641942395351412233546746461491743013475866037338176532604557402925472279360288926189385899695656876783017186873988763887625972743976062813263446647679413679726184933395074665820441677989806604203937116666336696282569349097348391155869004856032512219241534268522369316036765491047702733521540143168338872968405443296967684036073182435336224865433823598123544167514608340781166661858781733980624199254577853462678039039937557802759942957205281043775666979396838109341118959475766220191217535093638985465283078692370662512323684390235587636228324657161183714078807661162179517887972801841572019639084400269037450381192797170314489871815031319992111563908303017288012610642062005359240278277393918026391717720136125984776933980647063763022608885359937595079088789081791802195768033381968605120487107610874898411568740159953020639098138993261095538868264084012160831040525974539251576403732889086736948366404734622708560040891610782221943405179794550155347682966855320097501905581419914591124181501062255627411231571377358697194374130822027383843815940638571387913337592362330440453487233047240668784133333047898995255221468847973813568083995644533005222551320155267768895412770329278670827490041172076663112783638152343547681663121189086864991380236281775275946061211813342054791801619220346912760381900528012343973598270461499814511324618195658528232044658270082064934680251556511272822083811563192256509945201222666603226059396247019707668580396286975551115189973049085051758765306785758000660424066894170620303846785860257370634352599586885088679654004465187790208942935153217316750113738031466034642429489076322228133763299919641336502028627289268087560036613770607463575515079087982099722660130472907825746908175451952405573791313113170617323191598673971588373108168916968657704150695512947652386134815766967580364762005289060222744531744305498402863048850869557761528650326080941160688570698894762046478500884303973107412774191961697450517110329082815201273888663422631492147090220016940636504812047036016738602290671629816411198202268607961324739550057567564568204754619040423011062371367395995678940884705976859514505017241517746017351430990972615509378334720000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000

%time factorial_while(n)

CPU times: user 4.93 ms, sys: 0 ns, total: 4.93 ms
Wall time: 4.94 ms

422857792660554352220106420023358440539078667462664674884978240218135805270810820069089904787170638753708474665730068544587848606668381273633721089377278763127939036305846216064390447898698223987192970889621161265296832177550039924219683703146907264472878789790404754884162215226671928410969236910449565971736352948400223840381120644820230857671104502306174894755428309761781724040805324809927809328784055486199364548291211876258248802189173977900050213212598043639244626460770511358846595108675470585833924655225589035474435988347383178988034633008458631510209091509935653820010933047965742556741930917055172805200236075085991197635228755907902043369743123506916831211924495971556267407521462198986233088625998302859864857578749445963115286970886710046268423648178989905454690861391613218344174148807186234448114831209490361196546872767755617886828720269104814092456410341835975604276458161513178575901661071782544156980883359372729995603371371200471049437656291142488605335299499642300699972204918120100819059439140675053265004775533850899097945101551091486907004407119572336026243368132330218709287699196806656569752790422258267841561083376425781032629202687211070274681394351128601502326190649959171897364176378436491219709109840944514895358959103804176941956657834822071749105512752639148381172052604826965162642710094919393332661030104360530459117014557209584714353721948246686793467375904872268133410207860903657108806376616249749507413107077401682180585945526445171409277469230062697511346044174567946735828782261629584248675157379172942724178783105429858245117575511884506574424827574660800238588378492396247368761507015767725898321128632295537044902516387925127590841791744640466913531047347984464996154595542013996317357476301740036796192919942190762895445656261767041799538161133387312823511534152581309087915883638351664797225912944270653557142511737323807232632958121797916679692329687096923901003255574789055099807487061047230646195984955239657612208673866514171699307557691897902675157342075864796345338446835085965490727326321910504064289713096224505162064669468098869917122127404504020684923266241760132910227866687270305284709452526825496617772499645206699836925910690894082637401043498371591126455822280606361394115344316771769934353664284928294436414769615881993661388255577487709937004594753907845149034434521174560594039916268444697661821387470705325559577933196460996662145377564935474169708562389214773222865507182490430016186142192760452307670621142961767274704123616107220009743758647492753665149532164780849075146330071016691313420662882562618283865836983632108760710427516073348347788414796732427080410860761841281888307115098982135338406610652147087046874760995427473673509451553599769040367353385551052571682650317682405743993414862392331981432579182193321898940450865013610998098383993110996355981328001049731588596312131853801205046787642910669365600437305633431984879048998524701293300789344532868156679762880495532846386020133480265279836946393384995675049993707814746561543438930431384237878981847802886009971088695632988347711863122382785963653115132377931373647397429369411499028751972227999545182615488298951151926682112451355318472209990435355949887299922035062039816011086376236539782172380237846650673624510635034423187315338308212043804710999419227821039747552717416043890169723961305549371844836119803565896062025009093664399360172007383613354405094329072476518909502507724675841989412224659392163116352038147362479528539732089309533421910635702805576629720156556510767780805933453631121829561792887673002802450932122778852968418208261778476955644980385691275787372678040959158711733971103165232678060798127609246173504120182666874262805385275843979167609007743380748420751185119102921960339376280986753665085212869255321536787932521882574101866137054328973735862725370178558806639851350386944039604928258820180419178073649693885802597758398892014389747165465973510852605706234402069637065660129535734043582961473427275805630839510667375349259659518575646939723218275780003250593895303820539697558870511543073920827422440516299708739599768461206246629098112368012579891284802505094028916959765079395437191311379314427405135599630375642214527294341797246187597964074239147838993541565834716156858499036773056611353833367087548900413091981676330749041510337597307246885839246941715548295730750618505881581959528992660225626903439573313450666972952115230668696227920947779974336574472673471408928071411283888082693377378077293104110767513639476200610858040596019639058015761002337463869352228385801434957178125581445862930042479404065736859862007914604590255413929950088044710384758990326548097338166940500085452723713571394902463820308668541802838317527668064278489561005755859991718966786449154063570014497194249878920859731254275567514575206399118150736397483102490793841725653421894276769116598153430084637087769510295415136551734675054015239706042571746001089968440498845985477977905031632568489156557231006499726498721480800181770357701502983008879487243887718884416833034708723239505377642232944095773219137582371673924704216723002256883135779230394688900662466182532658490724406767024939579697217467485562998183149665611743997680482094166257463879660305171274925119226367615337524381656217330771650129520988754856467131862602387619964334867961514408328902061082833180891221325853682856469916007952105166960451695430614212305743006877217407155473217957577017595967640563812729153867513698712395570542350999228605975469962186195531354132139126436676900465429996811680550737866770665988027062972502001882845886145344368771455361304414465613369092862748276981946836480550952968681758714859972973082332924094777085275279923304892719633314751563311192746150389219290616780607901383451137066300684376267199885515143681266137319912103235469786756421210624899005553564022924345831264231038363416781719908354140411717740185950606674198348143345444247191436828225654380047860390575922417071802670646875454211626958746795398540784464654140381751149965273621123540880166990280149033225139460832668170930713868826549977374286127789417784752681328371818759103642140881783220739808059714203285309721443041845459183002833408705783138284973283761286182927136745161897366207237396132790944984014154408304074405393067540767126182547597130843470311389815695365971788564022750674237400323621850094765267521941901241387478279883426470873616812485384444012772521050072279315853096279121131160167772077952572613800240684421885453537121341902236379684012385255288607189967725694227433323948595075570839061877450159652184414998155476107548008054192318436948191732631430603548399790783307267636729090980772827355854348032260067472537097785464567761181807367424367391769863758072145859791485033700592994963793369100283444558089838054017635403737133019311293080958287612107380374800660269767842888358265737486556785868822015143046249655995760379768685318192365806469199584071845493606922169776137542662239658644989770921478134709127917460871630220821981434654245065731262683089579031012893360788644107230184805400373136014216229159146992019884148290014414312800903102107833305090238435726779416177246873411503598700003151092815700331081727415624680432977205070450456683898626301702989301145364477416856732512330376477881749036052572605520684370616116539755132541369303867783267208227323664249206432363089268768826650939691861683271739757479552993242406186992420363781929485368098035256331092448215269276219116259145886393677034653480367887126133367116968226450914997055448521259751870084720025674658752403932061045903070039438252019383102480929019684602472171298321628237994627125366359971898374425099120673688383738299653892030662843074547559074235345274029211606091346327684749522046010409575607348155101677203187580089224494752922031093841661588823584993931745149914395557357641584185479831702428523965451087525425464777294595230360946416541997797947136806344915998772409176443137371178542210740572121166868692153240490080384205921192622875440898261478908123698956367080804687628524499897440855677969456909042340530355943524640751677873953113928698614347227572144946891896093294375476741234907792754338349412323060078767610089949156126934038921148370217193387617823370358925817112869563450001367619897145400986643461922197676975930010555225198913002123021780831934330880446592954552165911855939202579781122952065357362914478404946474565003115498072056580360667380889572746464375428055819322299305089287806874537401327100274428317925355003451536693172112088227603942809788645727306979971285649576934354004030728440581746648376658498039958964243370183454151720285337810904113124462432903353964296651109482836884580127588701293156099225044518125460113274986014470437757313881001319276124676116614833528935557503106018449788994378274613854651708241613167681463911870000812845144341406739985430072772303758111613511094355614896323929750846383152930263582535361784837558519666949972251935515953807207838615142130284450051795239760968433198292598921623223582396390262548856855875458198371559008447860086745945709118128793228222051767509371866110013193625845223493949829511199280837860523506412769337548130609594264463425077601147334209139128541628183172262143783062962408149391997187528106367348876678481602342743230027158192404187686545826519361990687336892886715133840245486110982482004482721799496658712257174429044916781194824165631560303473833317665121218052780795958220298330611945164019413315550379662980215357680731124530585915969709973988055743550083279071844959752353594644354789680372126344509423070253995102864458237454677761013556916212309752286152053213998745673034127676503369636682306665552051562491132528926155863868503100849180920507680658265915276163719928694258350604859732273949286080260640627521341007801815105623787926212039424781833439433877206395801115809084190794320195178235740190546595990289617711776195270354051193727229722248442080440098750369411277686593022133010625031862085145076421052980508837197986052557750303949606158442838846866137510968441567309838079394349570013029265177957120625555851951313574029897589283475525334409858911400694449308432874005015554332587793895080241128538758725945136400838324944471346436826148195406004114845870234072926697740631325878634790667698266181501256117692275715291249164821702372884416357600996851100939411444677628186007072278522314941048564396255796808221289935799262208553889221164765220850367706476491496133789353761537391569177822237744837614120253342622508007300513473422771427333106345971803240244226950458090539326689103619381998838844036231795282435495362489670734155948067688515321073064476077859628627852283657244564306449096277517172656954238392941958409527253281659572534531428389629894005886539486824117113929627356938973482935854650278689437014798383826002058208853517073216288725214522205265969614962147884840129004507737252424605074339660818182960296019196314124998538422017695110361380561701016357743542531148669369994130940908368220071936435111978592782493491477052187226546109199597269439152400467901173602521030051886080337084840114810246351288263986170081804888380750203521448348740849154718714478857809574515499505005070789428842888410027877777455981132319940624176532148686316581736774410084063436959989519288310869124517866342559353458242589411390516469440377562665821577845936829909679754548350510473633770839151033854639602753486401635204633884342346714935641429160856846724874244782055113759168236472297793612971080302530934478115527737540458968990354808058309381267323593563098546564376209385371052808344607189076003388781618019853273759498566916704703448438363503416368325266403224174519476678140428319327482851882140344319384445475456765253419659194332132585432270070759038565239668227171300009189122050845185261514627937717597528852978637931711212529529443323757910072909001703558763798612480281463093944391916950129336315045285163539312868586427437296109446101235604877439863299611899755965996608749049271167685268675335991297583209089553296409523640116060078495005377892783750147344122123777907727134146647404489837589487675423294546899354220341669961366698976529978580795899055864050388507083137333076839766882463680992355219727241831735127646189112380485883115569477888101759708977682149644340317924443085170303692214137621194388641989508360339306459037361842937028710758321966607546113761076362543928614316242890754021082233620012309384737312220374269033838579928678572939434168287053763374091938184632261131740934278117918891642447513543478446040549455379834556163353815868441692054518698919434175386663900335756765603264363767906721626620330878425545157208117246381251512669846685887209013144861632560461019513371814585249988176629925142145014710206193190373671380347663431029705222414785030188275106347446241258707937339085095757724316735066885094208761536164440443755860160625837091300574162065273670941888667964570550744724714137001968165215954380698515999483361357521322106131884771926641942395351412233546746461491743013475866037338176532604557402925472279360288926189385899695656876783017186873988763887625972743976062813263446647679413679726184933395074665820441677989806604203937116666336696282569349097348391155869004856032512219241534268522369316036765491047702733521540143168338872968405443296967684036073182435336224865433823598123544167514608340781166661858781733980624199254577853462678039039937557802759942957205281043775666979396838109341118959475766220191217535093638985465283078692370662512323684390235587636228324657161183714078807661162179517887972801841572019639084400269037450381192797170314489871815031319992111563908303017288012610642062005359240278277393918026391717720136125984776933980647063763022608885359937595079088789081791802195768033381968605120487107610874898411568740159953020639098138993261095538868264084012160831040525974539251576403732889086736948366404734622708560040891610782221943405179794550155347682966855320097501905581419914591124181501062255627411231571377358697194374130822027383843815940638571387913337592362330440453487233047240668784133333047898995255221468847973813568083995644533005222551320155267768895412770329278670827490041172076663112783638152343547681663121189086864991380236281775275946061211813342054791801619220346912760381900528012343973598270461499814511324618195658528232044658270082064934680251556511272822083811563192256509945201222666603226059396247019707668580396286975551115189973049085051758765306785758000660424066894170620303846785860257370634352599586885088679654004465187790208942935153217316750113738031466034642429489076322228133763299919641336502028627289268087560036613770607463575515079087982099722660130472907825746908175451952405573791313113170617323191598673971588373108168916968657704150695512947652386134815766967580364762005289060222744531744305498402863048850869557761528650326080941160688570698894762046478500884303973107412774191961697450517110329082815201273888663422631492147090220016940636504812047036016738602290671629816411198202268607961324739550057567564568204754619040423011062371367395995678940884705976859514505017241517746017351430990972615509378334720000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000

%time factorial_for(n)

CPU times: user 4.82 ms, sys: 0 ns, total: 4.82 ms
Wall time: 4.84 ms

422857792660554352220106420023358440539078667462664674884978240218135805270810820069089904787170638753708474665730068544587848606668381273633721089377278763127939036305846216064390447898698223987192970889621161265296832177550039924219683703146907264472878789790404754884162215226671928410969236910449565971736352948400223840381120644820230857671104502306174894755428309761781724040805324809927809328784055486199364548291211876258248802189173977900050213212598043639244626460770511358846595108675470585833924655225589035474435988347383178988034633008458631510209091509935653820010933047965742556741930917055172805200236075085991197635228755907902043369743123506916831211924495971556267407521462198986233088625998302859864857578749445963115286970886710046268423648178989905454690861391613218344174148807186234448114831209490361196546872767755617886828720269104814092456410341835975604276458161513178575901661071782544156980883359372729995603371371200471049437656291142488605335299499642300699972204918120100819059439140675053265004775533850899097945101551091486907004407119572336026243368132330218709287699196806656569752790422258267841561083376425781032629202687211070274681394351128601502326190649959171897364176378436491219709109840944514895358959103804176941956657834822071749105512752639148381172052604826965162642710094919393332661030104360530459117014557209584714353721948246686793467375904872268133410207860903657108806376616249749507413107077401682180585945526445171409277469230062697511346044174567946735828782261629584248675157379172942724178783105429858245117575511884506574424827574660800238588378492396247368761507015767725898321128632295537044902516387925127590841791744640466913531047347984464996154595542013996317357476301740036796192919942190762895445656261767041799538161133387312823511534152581309087915883638351664797225912944270653557142511737323807232632958121797916679692329687096923901003255574789055099807487061047230646195984955239657612208673866514171699307557691897902675157342075864796345338446835085965490727326321910504064289713096224505162064669468098869917122127404504020684923266241760132910227866687270305284709452526825496617772499645206699836925910690894082637401043498371591126455822280606361394115344316771769934353664284928294436414769615881993661388255577487709937004594753907845149034434521174560594039916268444697661821387470705325559577933196460996662145377564935474169708562389214773222865507182490430016186142192760452307670621142961767274704123616107220009743758647492753665149532164780849075146330071016691313420662882562618283865836983632108760710427516073348347788414796732427080410860761841281888307115098982135338406610652147087046874760995427473673509451553599769040367353385551052571682650317682405743993414862392331981432579182193321898940450865013610998098383993110996355981328001049731588596312131853801205046787642910669365600437305633431984879048998524701293300789344532868156679762880495532846386020133480265279836946393384995675049993707814746561543438930431384237878981847802886009971088695632988347711863122382785963653115132377931373647397429369411499028751972227999545182615488298951151926682112451355318472209990435355949887299922035062039816011086376236539782172380237846650673624510635034423187315338308212043804710999419227821039747552717416043890169723961305549371844836119803565896062025009093664399360172007383613354405094329072476518909502507724675841989412224659392163116352038147362479528539732089309533421910635702805576629720156556510767780805933453631121829561792887673002802450932122778852968418208261778476955644980385691275787372678040959158711733971103165232678060798127609246173504120182666874262805385275843979167609007743380748420751185119102921960339376280986753665085212869255321536787932521882574101866137054328973735862725370178558806639851350386944039604928258820180419178073649693885802597758398892014389747165465973510852605706234402069637065660129535734043582961473427275805630839510667375349259659518575646939723218275780003250593895303820539697558870511543073920827422440516299708739599768461206246629098112368012579891284802505094028916959765079395437191311379314427405135599630375642214527294341797246187597964074239147838993541565834716156858499036773056611353833367087548900413091981676330749041510337597307246885839246941715548295730750618505881581959528992660225626903439573313450666972952115230668696227920947779974336574472673471408928071411283888082693377378077293104110767513639476200610858040596019639058015761002337463869352228385801434957178125581445862930042479404065736859862007914604590255413929950088044710384758990326548097338166940500085452723713571394902463820308668541802838317527668064278489561005755859991718966786449154063570014497194249878920859731254275567514575206399118150736397483102490793841725653421894276769116598153430084637087769510295415136551734675054015239706042571746001089968440498845985477977905031632568489156557231006499726498721480800181770357701502983008879487243887718884416833034708723239505377642232944095773219137582371673924704216723002256883135779230394688900662466182532658490724406767024939579697217467485562998183149665611743997680482094166257463879660305171274925119226367615337524381656217330771650129520988754856467131862602387619964334867961514408328902061082833180891221325853682856469916007952105166960451695430614212305743006877217407155473217957577017595967640563812729153867513698712395570542350999228605975469962186195531354132139126436676900465429996811680550737866770665988027062972502001882845886145344368771455361304414465613369092862748276981946836480550952968681758714859972973082332924094777085275279923304892719633314751563311192746150389219290616780607901383451137066300684376267199885515143681266137319912103235469786756421210624899005553564022924345831264231038363416781719908354140411717740185950606674198348143345444247191436828225654380047860390575922417071802670646875454211626958746795398540784464654140381751149965273621123540880166990280149033225139460832668170930713868826549977374286127789417784752681328371818759103642140881783220739808059714203285309721443041845459183002833408705783138284973283761286182927136745161897366207237396132790944984014154408304074405393067540767126182547597130843470311389815695365971788564022750674237400323621850094765267521941901241387478279883426470873616812485384444012772521050072279315853096279121131160167772077952572613800240684421885453537121341902236379684012385255288607189967725694227433323948595075570839061877450159652184414998155476107548008054192318436948191732631430603548399790783307267636729090980772827355854348032260067472537097785464567761181807367424367391769863758072145859791485033700592994963793369100283444558089838054017635403737133019311293080958287612107380374800660269767842888358265737486556785868822015143046249655995760379768685318192365806469199584071845493606922169776137542662239658644989770921478134709127917460871630220821981434654245065731262683089579031012893360788644107230184805400373136014216229159146992019884148290014414312800903102107833305090238435726779416177246873411503598700003151092815700331081727415624680432977205070450456683898626301702989301145364477416856732512330376477881749036052572605520684370616116539755132541369303867783267208227323664249206432363089268768826650939691861683271739757479552993242406186992420363781929485368098035256331092448215269276219116259145886393677034653480367887126133367116968226450914997055448521259751870084720025674658752403932061045903070039438252019383102480929019684602472171298321628237994627125366359971898374425099120673688383738299653892030662843074547559074235345274029211606091346327684749522046010409575607348155101677203187580089224494752922031093841661588823584993931745149914395557357641584185479831702428523965451087525425464777294595230360946416541997797947136806344915998772409176443137371178542210740572121166868692153240490080384205921192622875440898261478908123698956367080804687628524499897440855677969456909042340530355943524640751677873953113928698614347227572144946891896093294375476741234907792754338349412323060078767610089949156126934038921148370217193387617823370358925817112869563450001367619897145400986643461922197676975930010555225198913002123021780831934330880446592954552165911855939202579781122952065357362914478404946474565003115498072056580360667380889572746464375428055819322299305089287806874537401327100274428317925355003451536693172112088227603942809788645727306979971285649576934354004030728440581746648376658498039958964243370183454151720285337810904113124462432903353964296651109482836884580127588701293156099225044518125460113274986014470437757313881001319276124676116614833528935557503106018449788994378274613854651708241613167681463911870000812845144341406739985430072772303758111613511094355614896323929750846383152930263582535361784837558519666949972251935515953807207838615142130284450051795239760968433198292598921623223582396390262548856855875458198371559008447860086745945709118128793228222051767509371866110013193625845223493949829511199280837860523506412769337548130609594264463425077601147334209139128541628183172262143783062962408149391997187528106367348876678481602342743230027158192404187686545826519361990687336892886715133840245486110982482004482721799496658712257174429044916781194824165631560303473833317665121218052780795958220298330611945164019413315550379662980215357680731124530585915969709973988055743550083279071844959752353594644354789680372126344509423070253995102864458237454677761013556916212309752286152053213998745673034127676503369636682306665552051562491132528926155863868503100849180920507680658265915276163719928694258350604859732273949286080260640627521341007801815105623787926212039424781833439433877206395801115809084190794320195178235740190546595990289617711776195270354051193727229722248442080440098750369411277686593022133010625031862085145076421052980508837197986052557750303949606158442838846866137510968441567309838079394349570013029265177957120625555851951313574029897589283475525334409858911400694449308432874005015554332587793895080241128538758725945136400838324944471346436826148195406004114845870234072926697740631325878634790667698266181501256117692275715291249164821702372884416357600996851100939411444677628186007072278522314941048564396255796808221289935799262208553889221164765220850367706476491496133789353761537391569177822237744837614120253342622508007300513473422771427333106345971803240244226950458090539326689103619381998838844036231795282435495362489670734155948067688515321073064476077859628627852283657244564306449096277517172656954238392941958409527253281659572534531428389629894005886539486824117113929627356938973482935854650278689437014798383826002058208853517073216288725214522205265969614962147884840129004507737252424605074339660818182960296019196314124998538422017695110361380561701016357743542531148669369994130940908368220071936435111978592782493491477052187226546109199597269439152400467901173602521030051886080337084840114810246351288263986170081804888380750203521448348740849154718714478857809574515499505005070789428842888410027877777455981132319940624176532148686316581736774410084063436959989519288310869124517866342559353458242589411390516469440377562665821577845936829909679754548350510473633770839151033854639602753486401635204633884342346714935641429160856846724874244782055113759168236472297793612971080302530934478115527737540458968990354808058309381267323593563098546564376209385371052808344607189076003388781618019853273759498566916704703448438363503416368325266403224174519476678140428319327482851882140344319384445475456765253419659194332132585432270070759038565239668227171300009189122050845185261514627937717597528852978637931711212529529443323757910072909001703558763798612480281463093944391916950129336315045285163539312868586427437296109446101235604877439863299611899755965996608749049271167685268675335991297583209089553296409523640116060078495005377892783750147344122123777907727134146647404489837589487675423294546899354220341669961366698976529978580795899055864050388507083137333076839766882463680992355219727241831735127646189112380485883115569477888101759708977682149644340317924443085170303692214137621194388641989508360339306459037361842937028710758321966607546113761076362543928614316242890754021082233620012309384737312220374269033838579928678572939434168287053763374091938184632261131740934278117918891642447513543478446040549455379834556163353815868441692054518698919434175386663900335756765603264363767906721626620330878425545157208117246381251512669846685887209013144861632560461019513371814585249988176629925142145014710206193190373671380347663431029705222414785030188275106347446241258707937339085095757724316735066885094208761536164440443755860160625837091300574162065273670941888667964570550744724714137001968165215954380698515999483361357521322106131884771926641942395351412233546746461491743013475866037338176532604557402925472279360288926189385899695656876783017186873988763887625972743976062813263446647679413679726184933395074665820441677989806604203937116666336696282569349097348391155869004856032512219241534268522369316036765491047702733521540143168338872968405443296967684036073182435336224865433823598123544167514608340781166661858781733980624199254577853462678039039937557802759942957205281043775666979396838109341118959475766220191217535093638985465283078692370662512323684390235587636228324657161183714078807661162179517887972801841572019639084400269037450381192797170314489871815031319992111563908303017288012610642062005359240278277393918026391717720136125984776933980647063763022608885359937595079088789081791802195768033381968605120487107610874898411568740159953020639098138993261095538868264084012160831040525974539251576403732889086736948366404734622708560040891610782221943405179794550155347682966855320097501905581419914591124181501062255627411231571377358697194374130822027383843815940638571387913337592362330440453487233047240668784133333047898995255221468847973813568083995644533005222551320155267768895412770329278670827490041172076663112783638152343547681663121189086864991380236281775275946061211813342054791801619220346912760381900528012343973598270461499814511324618195658528232044658270082064934680251556511272822083811563192256509945201222666603226059396247019707668580396286975551115189973049085051758765306785758000660424066894170620303846785860257370634352599586885088679654004465187790208942935153217316750113738031466034642429489076322228133763299919641336502028627289268087560036613770607463575515079087982099722660130472907825746908175451952405573791313113170617323191598673971588373108168916968657704150695512947652386134815766967580364762005289060222744531744305498402863048850869557761528650326080941160688570698894762046478500884303973107412774191961697450517110329082815201273888663422631492147090220016940636504812047036016738602290671629816411198202268607961324739550057567564568204754619040423011062371367395995678940884705976859514505017241517746017351430990972615509378334720000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000

Example: The Fibonacci sequence

Fib(0) = 0 (base case 1)

Fib(1) = 1 (base case 2)

For all integers n > 1, Fib(n) = Fib(n − 1) + Fib(n − 2)

def Fibonacci(n):
    "Compute a Fibonacci Sequence using recursion"

    # If n is negative
    if n < 0:
        print("Incorrect input. Value must be 0 or greater.")

    # If n is 0
    elif n == 0:
        return 0

    # If n is 1 or 2
    elif n == 1 or n == 2:
        return 1

    else:
        return Fibonacci(n - 1) + Fibonacci(n - 2)

n = 9

Fibonacci(9)

for n in range(100):
    if n > 0: print(", ", end="")
    print(Fibonacci(n), end="")

As a for loop

def fibber(r:int = 10):
    """
    Computes a Fibonacci Sequence using a for loop. 
    Parameter r must be in integer > 3. Defaults to 10.
    Returns a string as a comma-limited series.
    """
    seq = [1,1,2] 
    kernel = lambda x, i: x[i-1] + x[i-2]
    for n in range(3, r):
        seq.append(seq[n-1] + seq[n-2])
    return ', '.join([str(x) for x in seq])

fibber(20)

Aside: A General Sequence Function

Recursive functions are often used to produce mathematical sequences, but since they have limits on depth, they are of limited use for this purpose.

Here is a function that can combine many sequences using two sequence parameters: * The initial state of the sequence, represented as the list seq. * For example, in the Fibonacci sequence, seq is [1, 1, 2] * The function to apply to the sequence at each iteration, represneted as a lambda function with the arguments x and i for the the sequence list seq and the iteration number respectively. * For example, in the Fibonacci sequence the kernel function is lambda x, i: x[i-1] + x[i-2]

##| tags: []
def sequencer(n:int = 10, seq=[1, 1, 2], kernel=lambda x, i: x[i-1] + x[i-2]):
    """
    Computes a Sequence using a for loop. 
    
    Parameter n in integer which must be > 3. Defaults to 10.
    Parameter seq is as list in the initial state of the sequence. Must have at least one value. Defaults to Fibonacci [1,1,2]
    Parameter kernel is the kernel function applied to the series at each iteration. x stands for the seq list, i to the iteration number. Defaults to lambda x, i: x[i-1] + x[i-2]
    
    Returns a string as a comma-limited series.
    """
    
    for i in range(len(seq), n): seq.append(kernel(seq, i))
    return ', '.join([str(x) for x in seq])

n = 8

%time sequencer(n, [0], lambda x, i: i)

The series of positive integers

sequencer(n, [1], lambda x, i: x[i-1] + 1)

The series of even numbers

sequencer(n, [2], lambda x, i: x[i-1] + 2)

The series of odd numbers

sequencer(n, [1], lambda x, i: x[i-1] + 2)

The series of Fibonacci numbers

sequencer(n, [1,1,2], lambda x, i: x[i-1] + x[i-2])

The series of Squares

sequencer(n, [2], lambda x, i: x[i-1]**2)