FIBONACCI

[Xlambda]

FIBONACCI !! recursive !! calculates the nth term of the famous Fibonacci sequence . This is more of a fun quest than a useful function. ( did this on a YT channel also, had fun with Rico about this).
This reflects how limited our computers are. You will be surprised. This time is not about recursion "iterations" limit, is about the limit of the number of calculations that a computer can perform.
Open a new workbook, define the lambda FIBONACCI. Very important, first, try values of n less then 20!! . Call FIBONACC(15) ...(19) ..etc. and increase it bit by bit.
For what number "n" do you think that your computer will start struggling giving you the answer?? Share with us here, if you please, how long it took your computer to calculate FIBONACCI(35)..(38) and (40). Eventually what type of computer you are using. Processor ,RAM etc. After all, 40 is a very small number.? Give it a try!!
Note: Fibonacci sequence is this one : 1,1,2,3,5,8,13,21,34,55,89,144....the value of n term is the sum of the previous 2 terms fib( n)=fib(n-1) + fib(n-2), and fib(1)=1,fib(2)=1
Fortunately , Fibonacci has also an explicit solution (non recursive), you can define another lambda to check the results FIB( n)=LAMBDA( n,LET(a,SQRT(5),x,((1+a)/2)^n,y,((1-a)/2)^n,1/a*(x-y)))

=LAMBDA(n,IF(n<3,1,FIBBONACCI(n-1)+FIBONACCI(n-2)))

LAMBDA 6.0.xlsx
	A	B	C	D	E	F	G	H
1	FIBONACCI		recursive	ckeck(non recursive)
2		n value	=FIBONACCI(B3)	=FIB(B3)	time(can use phone's stopwatch)
3		15	610	610	instant
4		18	2584	2584	instant
5		20	6765	6765	instant
6		30	832040	832040	1s
7		35	9227465	9227465	?s	share this
8		38	39088169	39088169	?s	values
9		40	102334155	102334155	?s	with us ?
10
FIBONACCI post

Cell Formulas
Range	Formula
C2:D2	C2	=FORMULATEXT(C3)
C3:C9	C3	=FIBONACCI(B3)
D3:D9	D3	=FIB(B3)

 

[Xlambda]

Lots of chatter here, but no mention that the LAMBDA posted has a typo. Posted is
"=LAMBDA(n,IF(n<3,1,FIBBONACCI(n-1)+FIBONACCI(n-2)))", but for any value 3 or more it just generates a #NAME! error. Should be

=LAMBDA(n,IF(n<3,1,FIBONACCI(n-1)+FIBONACCI(n-2)))

Just sayin'.

Indeed, a good point, and the explanation is quite simple. The formula is so short that everybody typed it and they did it correctly, they did not copy mine, like you did. And when we type typos are possible.
Anyhow, in the minisheet all the functions are fine.
Hope that Fibonacci(10000) calculated here FIBONACCI has all the digits ok.

 

[Xlambda]

Momentous Day Today!! Happy 25th anniversary mrexcel.com !!
Congratulations to @MrExcel, to @smozgur and to all who made this amazing cutting-edge Best in The World, One of a kind, Second to None, Excel Laboratory, possible.
As a late "participant" I have realized that this is not only a place where problems get solved, it's more than that, it's about passion. And I am proud and grateful to be part of it among all the other pasionate people.
I thought that the best way to celebrate is to post something special to me, from my unpublished studies, something cool enough for this special day, something that reflects the passion, and here it is, an updated way to calculate Fibonacci nr. , replacing recursive design with an iterative reduce with array accumulator, simpler and lot faster than before.
Main function is a function designed to add exceptionally large numbers (numbers formatted as text, to keep the precision)
LADD(a) Large nr. ADDing
a: any array of large text numbers or regular numbers, positive integers
(The study has functions for all operations with large nr. : adding, multiplication, subtraction, dividing, sorting, comparing etc...but for FIB LADD is enough

=LAMBDA(a,
    LET(
        m, LZ(a, 1),
        n, COLUMNS(m),
        d, REDUCE(
            {0, ""},
            SEQUENCE(n, , n, -1),
            LAMBDA(v, i, LET(x, INDEX(m, , i), s, SUM(x) + INDEX(v, 1), g, LEN(s), l, IF(g = 1, 0, LEFT(s, g - 1)), r, RIGHT(s), HSTACK(l, r & INDEX(v, 2))))
        ),
        XZ(CONCAT(d))
    )
)

calls 2 tiny tool functions LZ(a,[e]) Leading 0's and XZ(a) eXclude leading 0's
a: array
e: expand arg., if omitted strings, if 1 expanded with MID

=LAMBDA(a, [e], LET(l, LEN(a), n, MAX(l), z, REPT(0, n - l) & a, m, --MID(TOCOL(z), SEQUENCE(, n), 1), IF(e, m, z)))

=LAMBDA(a, MAP(a, LAMBDA(x, IFERROR(REPLACE(x, 1, MIN(TOCOL(SEARCH(VSTACK("-", SEQUENCE(9)), x), 2)) - 1, ""), "0"))))

and FIB(n ) that calls LADD(a)

=LAMBDA(n,
    TAKE(REDUCE({0; 1}, SEQUENCE(n - 1), LAMBDA(v, i, VSTACK(INDEX(v, 2), LADD(VSTACK(INDEX(v, 1), INDEX(v, 2)))))), -1)
)

Other functions on mini-sheet
WS(s,c) Wrap String s: string to wrap; c: nr. chars /row

=LAMBDA(s, c, MID(s, SEQUENCE(ROUNDUP(LEN(s) / c, 0), , , c), c))

25 years.xlsx
	A	B	C	D	E	F	G	H	I	J	K	L	M	N	O	P	Q	R	S	T	U	V	W	X	Y
1		LZ(a,[e]) and XZ(a) LADD(a) funcionality
2
3		98765399876543456543	123
4		0
5
6		=LZ(B3:C4)			=LZ(B3:C4,1)
7		98765399876543456543	00000000000000000123		9	8	7	6	5	3	9	9	8	7	6	5	4	3	4	5	6	5	4	3
8		00000000000000000000	00000000000000000000		0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	1	2	3
9					0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
10					0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
11
12		=XZ(B7#)
13		98765399876543456543	123
14		0	0
15
16		=LADD(B7#)
17		98765399876543456666
18					Adiing 500000 numbers of 20 digits each
19
20		=IF(SEQUENCE(250000,2),REPT(9,20))			=LADD(B21#)
21		99999999999999999999	99999999999999999999		49999999999999999999500000
22		99999999999999999999	99999999999999999999
23		99999999999999999999	99999999999999999999
Sheet1

Cell Formulas
Range	Formula
E6,E20,B20,B16,B12,B6	E6	=FORMULATEXT(E7)
B7:C8	B7	=LZ(B3:C4)
E7:X10	E7	=LZ(B3:C4,1)
B13:C14	B13	=XZ(B7#)
B17	B17	=LADD(B7#)
B21:C250020	B21	=IF(SEQUENCE(250000,2),REPT(9,20))
E21	E21	=LADD(B21#)
Dynamic array formulas.

 

[Xlambda]

If with previous method FIB(5000) took 18min now is less than 1min to calculate!!!
wolfram alpha link to check values fib(10000) - Wolfram|Alpha

Book1
	B	C	D
1
2	fib nr
3	1000		43466557686937456435688527675040625802564660517371780402481729089536555417949051890403879840079255169295922593080322634775209689623239873322471161642996440906533187938298969649928516003704476137795166849228875
4
5
6	2000		4224696333392304878706725602341482782579852840250681098010280137314308584370130707224123599639141511088446087538909603607640194711643596029271983312598737326253555802606991585915229492453904998722256795316982874482472992263901833716778060607011615497886719879858311468870876264597369086722884023654422295243347964480139515349562972087652656069529806499841977448720155612802665404554171717881930324025204312082516817125
7
8
9			422469633339230487870672560234148278257985284025068109801028013731430858437013070722412359963914151108844608753890960360
10			764019471164359602927198331259873732625355580260699158591522949245390499872225679531698287448247299226390183371677806060
11			701161549788671987985831146887087626459736908672288402365442229524334796448013951534956297208765265606952980649984197744
12			8720155612802665404554171717881930324025204312082516817125
13
14
15	3000		410615886307971260333568378719267105220125108637369252408885430926905584274113403731330491660850044560830036835706942274588569362145476502674373045446852160486606292497360503469773453733196887405847255290082049086907512622059054542195889758031109222670849274793859539133318371244795543147611073276240066737934085191731810993201706776838934766764778739502174470268627820918553842225858306408301661862900358266857238210235802504351951472997919676524004784236376453347268364152648346245840573214241419937917242918602639810097866942392015404620153818671425739835074851396421139982713640679581178458198658692285968043243656709796000
16
17
18			410615886307971260333568378719267105220125108637369252408885430926905584274113403731330491660850044560830036835706942274
19			588569362145476502674373045446852160486606292497360503469773453733196887405847255290082049086907512622059054542195889758
20			031109222670849274793859539133318371244795543147611073276240066737934085191731810993201706776838934766764778739502174470
21			268627820918553842225858306408301661862900358266857238210235802504351951472997919676524004784236376453347268364152648346
22			245840573214241419937917242918602639810097866942392015404620153818671425739835074851396421139982713640679581178458198658
23			692285968043243656709796000
24
25
26	4000		39909473435004422792081248094960912600792570982820257852628876326523051818641373433549136769424132442293969306537520118273879628025443235370362250955435654171592897966790864814458223141914272590897468472180370639695334449662650312874735560926298246249404168309064214351044459077749425236777660809226095151852052781352975449482565838369809183771787439660825140502824343131911711296392457138867486593923544177893735428602238212249156564631452507658603400012003685322984838488962351492632577755354452904049241294565662519417235020049873873878602731379207893212335423484873469083054556329894167262818692599815209582517277965059068235543139459375028276851221435815957374273143824422909416395375178739268544368126894240979135322176080374780998010657710775625856041594078495411724236560242597759185543824798332467919613598667003025993715274875
27
28
29			399094734350044227920812480949609126007925709828202578526288763265230518186413734335491367694241324422939693065375201182
30			738796280254432353703622509554356541715928979667908648144582231419142725908974684721803706396953344496626503128747355609
31			262982462494041683090642143510444590777494252367776608092260951518520527813529754494825658383698091837717874396608251405
32			028243431319117112963924571388674865939235441778937354286022382122491565646314525076586034000120036853229848384889623514
33			926325777553544529040492412945656625194172350200498738738786027313792078932123354234848734690830545563298941672628186925
34			998152095825172779650590682355431394593750282768512214358159573742731438244229094163953751787392685443681268942409791353
35			22176080374780998010657710775625856041594078495411724236560242597759185543824798332467919613598667003025993715274875
36
37
38	5000		3878968454388325633701916308325905312082127714646245106160597214895550139044037097010822916462210669479293452858882973813483102008954982940361430156911478938364216563944106910214505634133706558656238254656700712525929903854933813928836378347518908762970712033337052923107693008518093849801803847813996748881765554653788291644268912980384613778969021502293082475666346224923071883324803280375039130352903304505842701147635242270210934637699104006714174883298422891491273104054328753298044273676822977244987749874555691907703880637046832794811358973739993110106219308149018570815397854379195305617510761053075688783766033667355445258844886241619210553457493675897849027988234351023599844663934853256411952221859563060475364645470760330902420806382584929156452876291575759142343809142302917491088984155209854432486594079793571316841692868039545309545388698114665082066862897420639323438488465240988742395873801976993820317174208932265468879364002630797780058759129671389634214252579116872755600360311370547754724604639987588046985178408674382863125
39
40
41			387896845438832563370191630832590531208212771464624510616059721489555013904403709701082291646221066947929345285888297381
42			348310200895498294036143015691147893836421656394410691021450563413370655865623825465670071252592990385493381392883637834
43			751890876297071203333705292310769300851809384980180384781399674888176555465378829164426891298038461377896902150229308247
44			566634622492307188332480328037503913035290330450584270114763524227021093463769910400671417488329842289149127310405432875
45			329804427367682297724498774987455569190770388063704683279481135897373999311010621930814901857081539785437919530561751076
46			105307568878376603366735544525884488624161921055345749367589784902798823435102359984466393485325641195222185956306047536
47			464547076033090242080638258492915645287629157575914234380914230291749108898415520985443248659407979357131684169286803954
48			530954538869811466508206686289742063932343848846524098874239587380197699382031717420893226546887936400263079778005875912
49			9671389634214252579116872755600360311370547754724604639987588046985178408674382863125
50
51
Sheet1

Cell Formulas
Range	Formula
D3	D3	=FIB(1000)
D6	D6	=FIB(2000)
D9:D12,D41:D49,D29:D35,D18:D23	D9	=WS(D6,120)
D15	D15	=FIB(3000)
D26	D26	=FIB(4000)
D38	D38	=FIB(5000)
Dynamic array formulas.

Book1
	B	C	D
1
2	fib nr
3	10000		33644764876431783266621612005107543310302148460680063906564769974680081442166662368155595513633734025582065332680836159373734790483865268263040892463056431887354544369559827491606602099884183933864652731300088830269235673613135117579297437854413752130520504347701602264758318906527890855154366159582987279682987510631200575428783453215515103870818298969791613127856265033195487140214287532698187962046936097879900350962302291026368131493195275630227837628441540360584402572114334961180023091208287046088923962328835461505776583271252546093591128203925285393434620904245248929403901706233888991085841065183173360437470737908552631764325733993712871937587746897479926305837065742830161637408969178426378624212835258112820516370298089332099905707920064367426202389783111470054074998459250360633560933883831923386783056136435351892133279732908133732642652633989763922723407882928177953580570993691049175470808931841056146322338217465637321248226383092103297701648054726243842374862411453093812206564914032751086643394517512161526545361333111314042436854805106765843493523836959653428071768775328348234345557366719731392746273629108210679280784718035329131176778924659089938635459327894523777674406192240337638674004021330343297496902028328145933418826817683893072003634795623117103101291953169794607632737589253530772552375943788434504067715555779056450443016640119462580972216729758615026968443146952034614932291105970676243268515992834709891284706740862008587135016260312071903172086094081298321581077282076353186624611278245537208532365305775956430072517744315051539600905168603220349163222640885248852433158051534849622434848299380905070483482449327453732624567755879089187190803662058009594743150052402532709746995318770724376825907419939632265984147498193609285223945039707165443156421328157688908058783183404917434556270520223564846495196112460268313970975069382648706613264507665074611512677522748621598642530711298441182622661057163515069260029861704945425047491378115154139941550671256271197133252763631939606902895650288268608362241082050562430701794976171121233066073310059947366875
4
5
6			336447648764317832666216120051075433103021484606800639065647699746800814421666623681555955136337340255820653326808361593
7			737347904838652682630408924630564318873545443695598274916066020998841839338646527313000888302692356736131351175792974378
8			544137521305205043477016022647583189065278908551543661595829872796829875106312005754287834532155151038708182989697916131
9			278562650331954871402142875326981879620469360978799003509623022910263681314931952756302278376284415403605844025721143349
10			611800230912082870460889239623288354615057765832712525460935911282039252853934346209042452489294039017062338889910858410
11			651831733604374707379085526317643257339937128719375877468974799263058370657428301616374089691784263786242128352581128205
12			163702980893320999057079200643674262023897831114700540749984592503606335609338838319233867830561364353518921332797329081
13			337326426526339897639227234078829281779535805709936910491754708089318410561463223382174656373212482263830921032977016480
14			547262438423748624114530938122065649140327510866433945175121615265453613331113140424368548051067658434935238369596534280
15			717687753283482343455573667197313927462736291082106792807847180353291311767789246590899386354593278945237776744061922403
16			376386740040213303432974969020283281459334188268176838930720036347956231171031012919531697946076327375892535307725523759
17			437884345040677155557790564504430166401194625809722167297586150269684431469520346149322911059706762432685159928347098912
18			847067408620085871350162603120719031720860940812983215810772820763531866246112782455372085323653057759564300725177443150
19			515396009051686032203491632226408852488524331580515348496224348482993809050704834824493274537326245677558790891871908036
20			620580095947431500524025327097469953187707243768259074199396322659841474981936092852239450397071654431564213281576889080
21			587831834049174345562705202235648464951961124602683139709750693826487066132645076650746115126775227486215986425307112984
22			411826226610571635150692600298617049454250474913781151541399415506712562711971332527636319396069028956502882686083622410
23			82050562430701794976171121233066073310059947366875
24
Sheet1

Cell Formulas
Range	Formula
D3	D3	=FIB(10000)
D6:D23	D6	=WS(D3,120)
Dynamic array formulas.

 

[Xlambda]

fib(20000) - Wolfram|Alpha

Wolfram|Alpha brings expert-level knowledge and capabilities to the broadest possible range of people—spanning all professions and education levels.

www.wolframalpha.com

25 years.xlsx
	B	C	D
1
2	fib nr
3	20000		2531162323732361242240155003520607291766356485802485278951929841991312781760541315230153423463758831637443488219211037689033673531462742885329724071555187618026931630449193158922771331642302030331971098689235780843478258502779200293635651897483309686042860996364443514558772156043691404155819572984971754278513112487985892718229593329483578531419148805380281624260900362993556916638613939977074685016188258584312329139526393558096840812970422952418558991855772306882442574855589237165219912238201311184749075137322987656049866305366913734924425822681338966507463855180236283582409861199212323835947891143765414913345008456022009455704210891637791911265475167769704477334859109822590053774932978465651023851447920601310106288957894301592502061560528131203072778677491443420921822590709910448617329156135355464620891788459566081572824889514296350670950824208245170667601726417091127999999941149913010424532046881958285409468463211897582215075436515584016297874572183907949257286261608612401379639484713101138120404671732190451327881433201025184027541696124114463488665359385870910331476156665889459832092710304159637019707297988417848767011085425271875588008671422491434005115288334343837778792282383576736341414410248994081564830202363820504190074504566612515965134665683289356188727549463732830075811851574961558669278847363279870595320099844676879457196432535973357128305390290471349480258751812890314779723508104229525161740643984423978659638233074463100366500571977234508464710078102581304823235436518145074482824812996511614161933313389889630935320139507075992100561077534028207257574257706278201308302642634678112591091843082665721697117838726431766741158743554298864560993255547608496686850185804659790217122426535133253371422250684486113457341827911625517128815447325958547912113242367201990672230681308819195941016156001961954700241576553750737681552256845421159386858399433450045903975167084252876848848085910156941603293424067793097271128806817514906531652407763118308162377033463203514657531210413149191213595455280387631030665594589183601575340027172997222489081631144728873621805528648768511368948639522975539046995395707688938978847084621586473529546678958226255042389998718141303055036060772003887773038422366913820397748550793178167220193346017430024134496141145991896227741842515718997898627269918236920453493946658273870473264523119133765447653295022886429174942653014656521909469613184983671431465934965489425515981067546087342348350724207583544436107294087637975025147846254526938442435644928231027868701394819091132912397475713787593612758364812687556725146456646878912169274219209708166678668152184941578590201953144030519381922273252666652671717526318606676754556170379350956342095455612780202199922615392785572481747913435560866995432578680971243966868110016581395696310922519803685837460795358384618017215468122880442252343684547233668502313239328352671318130604247460452134121833305284398726438573787798499612760939462427922917659263046333084007208056631996856315539698234022953452211505675629153637867252695056925345220084020071611220575700841268302638995272842160994219632684575364180160991884885091858259996299627148614456696661412745040519981575543804847463997422326563897043803732970397488471644906183310144691243649149542394691524972023935190633672827306116525712882959108434211652465621144702015336657459532134026915214509960877430595844287585350290234547564574848753110281101545931547225811763441710217452979668178025286460158324658852904105792472468108996135476637212057508192176910900422826969523438985332067597093454021924077101784215936539638808624420121459718286059401823614213214326004270471752802725625810953787713898846144256909835116371235019527013180204030167601567064268573820697948868982630904164685161783088076506964317303709708574052747204405282785965604677674192569851918643651835755242670293612851920696732320545562286110332140065912751551110134916256237884844001366366654055079721985816714803952429301558096968202261698837096090377863017797020488044826628817462866854321356787305635653577619877987998113667928954840972022833505708587561902023411398915823487627297968947621416912816367516125096563705174220460639857683971213093125
4
5	len
6	4180		253116232373236124224015500352060729176635648580248527895192984199131278176054131523015342346375883163744348821921103768
7			903367353146274288532972407155518761802693163044919315892277133164230203033197109868923578084347825850277920029363565189
8			748330968604286099636444351455877215604369140415581957298497175427851311248798589271822959332948357853141914880538028162
9			426090036299355691663861393997707468501618825858431232913952639355809684081297042295241855899185577230688244257485558923
10			716521991223820131118474907513732298765604986630536691373492442582268133896650746385518023628358240986119921232383594789
11			114376541491334500845602200945570421089163779191126547516776970447733485910982259005377493297846565102385144792060131010
12			628895789430159250206156052813120307277867749144342092182259070991044861732915613535546462089178845956608157282488951429
13			635067095082420824517066760172641709112799999994114991301042453204688195828540946846321189758221507543651558401629787457
14			218390794925728626160861240137963948471310113812040467173219045132788143320102518402754169612411446348866535938587091033
15			147615666588945983209271030415963701970729798841784876701108542527187558800867142249143400511528833434383777879228238357
16			673634141441024899408156483020236382050419007450456661251596513466568328935618872754946373283007581185157496155866927884
17			736327987059532009984467687945719643253597335712830539029047134948025875181289031477972350810422952516174064398442397865
18			963823307446310036650057197723450846471007810258130482323543651814507448282481299651161416193331338988963093532013950707
19			599210056107753402820725757425770627820130830264263467811259109184308266572169711783872643176674115874355429886456099325
20			554760849668685018580465979021712242653513325337142225068448611345734182791162551712881544732595854791211324236720199067
21			223068130881919594101615600196195470024157655375073768155225684542115938685839943345004590397516708425287684884808591015
22			694160329342406779309727112880681751490653165240776311830816237703346320351465753121041314919121359545528038763103066559
23			458918360157534002717299722248908163114472887362180552864876851136894863952297553904699539570768893897884708462158647352
24			954667895822625504238999871814130305503606077200388777303842236691382039774855079317816722019334601743002413449614114599
25			189622774184251571899789862726991823692045349394665827387047326452311913376544765329502288642917494265301465652190946961
26			318498367143146593496548942551598106754608734234835072420758354443610729408763797502514784625452693844243564492823102786
27			870139481909113291239747571378759361275836481268755672514645664687891216927421920970816667866815218494157859020195314403
28			051938192227325266665267171752631860667675455617037935095634209545561278020219992261539278557248174791343556086699543257
29			868097124396686811001658139569631092251980368583746079535838461801721546812288044225234368454723366850231323932835267131
30			813060424746045213412183330528439872643857378779849961276093946242792291765926304633308400720805663199685631553969823402
31			295345221150567562915363786725269505692534522008402007161122057570084126830263899527284216099421963268457536418016099188
32			488509185825999629962714861445669666141274504051998157554380484746399742232656389704380373297039748847164490618331014469
33			124364914954239469152497202393519063367282730611652571288295910843421165246562114470201533665745953213402691521450996087
34			743059584428758535029023454756457484875311028110154593154722581176344171021745297966817802528646015832465885290410579247
35			246810899613547663721205750819217691090042282696952343898533206759709345402192407710178421593653963880862442012145971828
36			605940182361421321432600427047175280272562581095378771389884614425690983511637123501952701318020403016760156706426857382
37			069794886898263090416468516178308807650696431730370970857405274720440528278596560467767419256985191864365183575524267029
38			361285192069673232054556228611033214006591275155111013491625623788484400136636665405507972198581671480395242930155809696
39			820226169883709609037786301779702048804482662881746286685432135678730563565357761987798799811366792895484097202283350570
40			8587561902023411398915823487627297968947621416912816367516125096563705174220460639857683971213093125
41
Sheet2

Cell Formulas
Range	Formula
D3	D3	=FIB(20000)
B6	B6	=LEN(D3)
D6:D40	D6	=WS(D3,120)
Dynamic array formulas.