nixpulvis
Fibonacci's Sequence 1436 words written by Nathan Lilienthal
The sequence 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946, 17711, 28657, 46368, 75025, 121393, 196418, 317811, 514229, 832040, 1346269, 2178309, 3524578, 5702887, 9227465, 14930352, 24157817, 39088169, 63245986, 102334155, \(\ldots\) is given by:
\[\begin{align} F_0 &= 0 \\ F_1 &= 1 \\ F_n &= F_{n-1} + F_{n-2} \end{align}\]Naturally, the sum of the first \(n\) numbers of the sequence can be found rather easily if you know \(F_{n+2}\).
\[\sum_{i=1}^n F_i = F_{n+2} - 1\]Simplest Recursive Solution, in Racket
Translating the definition of \(F_n\) directly into a recursive function yields a working implementation, but it’s not optimized for performance.
(define (fib n)
(cond [(or (= n 0)
(= n 1)) n]
[else (+ (fib (- n 1))
(fib (- n 2)))]))
(map fib '(0 1 2 3 4 5 6 7 8 9))
'(0 1 1 2 3 5 8 13 21 34)
; Takes too long...
(fib 100)
Tail Recursive, in Elixir
Tail calls are often optimized away from the stack, and as a result of the different algorithmic structure, runs much much faster on my machine.
defmodule Math do
def fib(n) do
fib_acc(0, 1, n)
end
def fib_acc(a, b, 0) do a end
def fib_acc(a, b, n) do
fib_acc(b, a+b, n-1)
end
end
Enum.map([0, 1, 2, 3, 4, 5, 6, 7, 8, 9], &Math.fib/1)
[0, 1, 1, 2, 3, 5, 8, 13, 21, 34]
# Very fast.
Math.fib(100)
354224848179261915075
# Still fast.
Math.fib(trunc(1.0e5))
...number too big to display
# Slow again for me...
Math.fib(trunc(1.0e6))
Iterative, in Rust
Like the tail resursive function, a simple while loop can acomplish the same
goal, and this time we’ll use an optimized big number library to allow us to
get past \(F_{10^5}\).
extern crate num_bigint;
extern crate num_traits;
fn fib(n: BigUint) -> BigUint {
let zero = BigUint::zero();
let one = BigUint::one();
let mut f = (n, zero.clone(), one.clone());
while f.0 > zero {
f = (f.0 - one.clone(), f.2.clone(), f.1 + f.2)
}
return f.1;
}
fn main() {
println!("{}", fib(BigUint::from(13 as u64)));
// Print some big numbers.
println!("{}", fib(BigUint::from(100 as u64)));
println!("{}", fib(BigUint::from(1.0e5 as u64)));
// Careful, while this program will likely work, the output may
// overwhelm the process reading STDOUT.
println!("{}", fib(BigUint::from(1.0e6 as u64)));
}
Closed Form Solution
A closed form solution can be found and is rather interesting, since \(\varphi\) is the golden ratio (a solution to the equation \(x^2 - x - 1 = 0\)).
\[\begin{align} \varphi &= \frac{1 + \sqrt 5}{2} \approx 1.6180 \\ \psi &= \frac{1 - \sqrt 5}{2} = -\frac{1}{\varphi} \\ F_n &= \frac{\varphi^n - \psi^n}{\varphi - \psi} = \frac{\varphi^n - \psi^n}{\sqrt 5} \end{align}\] \[= \frac{1}{\sqrt{5}} \left[\left(\frac{1 + \sqrt{5}}{2}\right)^n - \left(\frac{1 - \sqrt{5}}{2}\right)^n\right] \\\]Don’t belive it? Calculate any \(F_n\) and check, even try \(F_{10^{10}}\) yourself:
Input:
n = 10^10,
phi = (1 + sqrt(5))/2,
psi = - 1/phi,
(phi^n - psi^n) / sqrt(5)
Result:
\(1.413521976921693 \times 10^{2089876402}\)
mod closed {
pub fn fib(n: f64) -> f64 {
(1.0 / 5f64.sqrt()) * (((1.0 + 5f64.sqrt()) / 2.0).powf(n) -
((1.0 - 5f64.sqrt()) / 2.0).powf(n))
}
}
// This works great, however fails at n = 72.
fn main() {
for n in 0..100 {
assert_eq!(fib(BigUint::from(n as u64)),
BigUint::from(closed::fib(n as f64) as u64),
"testing {}", n);
}
}
Next, we should try and use BigDecimal types and see if we can get past
\(F_{72}\) without IEEE 754 floats.
Brainfuck
>++++++++++>+>+[
[+++++[>++++++++<-]>.<++++++[>--------<-]+<<<]>.>>[
[-]<[>+<-]>>[<<+>+>-]<[>+<-[>+<-[>+<-[>+<-[>+<-[>+<-
[>+<-[>+<-[>+<-[>[-]>+>+<<<-[>+<-]]]]]]]]]]]+>>>
]<<<
]
This program doesn't terminate; you will have to kill it.
Credit Daniel B Cristofani (cristofdathevanetdotcom)
If you have Rust installed, you can run and generate the Fibonacci sequence yourself with this brainfuck program.
git clone https://github.com/nixpulvis/brainfuck
cd brainfuck
cargo run fixtures/fib.b
0
1
1
2
3
5
8
13
21
34
55
89
144
233
377
610
987
1597
2584
4181
6765
10946
17711
28657
46368
75025
121393
196418
317811
514229
832040
1346269
2178309
3524578
5702887
9227465
14930352
24157817
39088169
63245986
102334155
165580141
267914296
433494437
701408733
1134903170
1836311903
2971215073
4807526976
7778742049
12586269025
20365011074
32951280099
53316291173
86267571272
139583862445
225851433717
365435296162
591286729879
956722026041
1548008755920
2504730781961
4052739537881
6557470319842
10610209857723
17167680177565
27777890035288
44945570212853
72723460248141
117669030460994
190392490709135
308061521170129
498454011879264
806515533049393
1304969544928657
2111485077978050
3416454622906707
5527939700884757
8944394323791464
14472334024676221
23416728348467685
37889062373143906
61305790721611591
99194853094755497
160500643816367088
259695496911122585
420196140727489673
679891637638612258
1100087778366101931
1779979416004714189
2880067194370816120
4660046610375530309
7540113804746346429
12200160415121876738
19740274219868223167
31940434634990099905
51680708854858323072
83621143489848422977
135301852344706746049
218922995834555169026
354224848179261915075
573147844013817084101
927372692193078999176
1500520536206896083277
2427893228399975082453
3928413764606871165730
6356306993006846248183
10284720757613717413913
16641027750620563662096
26925748508234281076009
43566776258854844738105
70492524767089125814114
114059301025943970552219
184551825793033096366333
298611126818977066918552
483162952612010163284885
781774079430987230203437
1264937032042997393488322
2046711111473984623691759
3311648143516982017180081
5358359254990966640871840
8670007398507948658051921
14028366653498915298923761
22698374052006863956975682
36726740705505779255899443
59425114757512643212875125
96151855463018422468774568
155576970220531065681649693
251728825683549488150424261
407305795904080553832073954
659034621587630041982498215
1066340417491710595814572169
1725375039079340637797070384
2791715456571051233611642553
4517090495650391871408712937
7308805952221443105020355490
11825896447871834976429068427
19134702400093278081449423917
30960598847965113057878492344
50095301248058391139327916261
81055900096023504197206408605
131151201344081895336534324866
212207101440105399533740733471
343358302784187294870275058337
555565404224292694404015791808
898923707008479989274290850145
1454489111232772683678306641953
2353412818241252672952597492098
3807901929474025356630904134051
6161314747715278029583501626149
9969216677189303386214405760200
16130531424904581415797907386349
26099748102093884802012313146549
42230279526998466217810220532898
68330027629092351019822533679447
110560307156090817237632754212345
178890334785183168257455287891792
289450641941273985495088042104137
468340976726457153752543329995929
757791618667731139247631372100066
1226132595394188293000174702095995
1983924214061919432247806074196061
3210056809456107725247980776292056
5193981023518027157495786850488117
8404037832974134882743767626780173
13598018856492162040239554477268290
22002056689466296922983322104048463
35600075545958458963222876581316753
57602132235424755886206198685365216
93202207781383214849429075266681969
150804340016807970735635273952047185
244006547798191185585064349218729154
394810887814999156320699623170776339
638817435613190341905763972389505493
1033628323428189498226463595560281832
1672445759041379840132227567949787325
2706074082469569338358691163510069157
4378519841510949178490918731459856482
7084593923980518516849609894969925639
11463113765491467695340528626429782121
18547707689471986212190138521399707760
30010821454963453907530667147829489881
48558529144435440119720805669229197641
78569350599398894027251472817058687522
127127879743834334146972278486287885163
205697230343233228174223751303346572685
332825110087067562321196029789634457848
538522340430300790495419781092981030533
871347450517368352816615810882615488381
1409869790947669143312035591975596518914
2281217241465037496128651402858212007295
3691087032412706639440686994833808526209
5972304273877744135569338397692020533504
9663391306290450775010025392525829059713
15635695580168194910579363790217849593217
25299086886458645685589389182743678652930
40934782466626840596168752972961528246147
66233869353085486281758142155705206899077
107168651819712326877926895128666735145224
173402521172797813159685037284371942044301
280571172992510140037611932413038677189525
453973694165307953197296969697410619233826
734544867157818093234908902110449296423351
1188518561323126046432205871807859915657177
1923063428480944139667114773918309212080528
3111581989804070186099320645726169127737705
5034645418285014325766435419644478339818233
8146227408089084511865756065370647467555938
13180872826374098837632191485015125807374171
21327100234463183349497947550385773274930109
34507973060837282187130139035400899082304280
55835073295300465536628086585786672357234389
90343046356137747723758225621187571439538669
146178119651438213260386312206974243796773058
236521166007575960984144537828161815236311727
382699285659014174244530850035136059033084785
619220451666590135228675387863297874269396512
1001919737325604309473206237898433933302481297
1621140188992194444701881625761731807571877809
2623059926317798754175087863660165740874359106
4244200115309993198876969489421897548446236915
6867260041627791953052057353082063289320596021
11111460156937785151929026842503960837766832936
17978720198565577104981084195586024127087428957
29090180355503362256910111038089984964854261893
47068900554068939361891195233676009091941690850
76159080909572301618801306271765994056795952743
123227981463641240980692501505442003148737643593
199387062373213542599493807777207997205533596336
322615043836854783580186309282650000354271239929
522002106210068326179680117059857997559804836265
844617150046923109759866426342507997914076076194
1366619256256991435939546543402365995473880912459
2211236406303914545699412969744873993387956988653
3577855662560905981638959513147239988861837901112
5789092068864820527338372482892113982249794889765
9366947731425726508977331996039353971111632790877
15156039800290547036315704478931467953361427680642
24522987531716273545293036474970821924473060471519
39679027332006820581608740953902289877834488152161
64202014863723094126901777428873111802307548623680
103881042195729914708510518382775401680142036775841
168083057059453008835412295811648513482449585399521
271964099255182923543922814194423915162591622175362
440047156314635932379335110006072428645041207574883
712011255569818855923257924200496343807632829750245
1152058411884454788302593034206568772452674037325128
1864069667454273644225850958407065116260306867075373
3016128079338728432528443992613633888712980904400501
4880197746793002076754294951020699004973287771475874
7896325826131730509282738943634332893686268675876375
12776523572924732586037033894655031898659556447352249
20672849399056463095319772838289364792345825123228624
33449372971981195681356806732944396691005381570580873
54122222371037658776676579571233761483351206693809497
87571595343018854458033386304178158174356588264390370
141693817714056513234709965875411919657707794958199867
229265413057075367692743352179590077832064383222590237
370959230771131880927453318055001997489772178180790104
600224643828207248620196670234592075321836561403380341
971183874599339129547649988289594072811608739584170445
1571408518427546378167846658524186148133445300987550786
2542592393026885507715496646813780220945054040571721231
4114000911454431885883343305337966369078499341559272017
6656593304481317393598839952151746590023553382130993248
10770594215935749279482183257489712959102052723690265265
17427187520417066673081023209641459549125606105821258513
28197781736352815952563206467131172508227658829511523778
45624969256769882625644229676772632057353264935332782291
73822750993122698578207436143903804565580923764844306069
119447720249892581203851665820676436622934188700177088360
193270471243015279782059101964580241188515112465021394429
312718191492907860985910767785256677811449301165198482789
505988662735923140767969869749836918999964413630219877218
818706854228831001753880637535093596811413714795418360007
1324695516964754142521850507284930515811378128425638237225
2143402371193585144275731144820024112622791843221056597232
3468097888158339286797581652104954628434169971646694834457
5611500259351924431073312796924978741056961814867751431689
9079598147510263717870894449029933369491131786514446266146
14691098406862188148944207245954912110548093601382197697835
23770696554372451866815101694984845480039225387896643963981
38461794961234640015759308940939757590587318989278841661816
62232491515607091882574410635924603070626544377175485625797
100694286476841731898333719576864360661213863366454327287613
162926777992448823780908130212788963731840407743629812913410
263621064469290555679241849789653324393054271110084140201023
426547842461739379460149980002442288124894678853713953114433
690168906931029935139391829792095612517948949963798093315456
1116716749392769314599541809794537900642843628817512046429889
1806885656323799249738933639586633513160792578781310139745345
2923602405716568564338475449381171413803636207598822186175234
4730488062040367814077409088967804926964428786380132325920579
7654090467756936378415884538348976340768064993978954512095813
12384578529797304192493293627316781267732493780359086838016392
20038668997554240570909178165665757608500558774338041350112205
32423247527351544763402471792982538876233052554697128188128597
52461916524905785334311649958648296484733611329035169538240802
84885164052257330097714121751630835360966663883732297726369399
137347080577163115432025771710279131845700275212767467264610201
222232244629420445529739893461909967206666939096499764990979600
359579325206583560961765665172189099052367214309267232255589801
581811569836004006491505558634099066259034153405766997246569401
941390895042587567453271223806288165311401367715034229502159202
1523202464878591573944776782440387231570435521120801226748728603
2464593359921179141398048006246675396881836888835835456250887805
3987795824799770715342824788687062628452272409956636682999616408
6452389184720949856740872794933738025334109298792472139250504213
10440185009520720572083697583620800653786381708749108822250120621
16892574194241670428824570378554538679120491007541580961500624834
27332759203762391000908267962175339332906872716290689783750745455
44225333398004061429732838340729878012027363723832270745251370289
71558092601766452430641106302905217344934236440122960529002115744