# Sylvester's sequence

Sylvester's sequence, OEIS A000058, is an integer sequence defined as follows:

Each member is the product of all previous members plus one. The first member of the sequence is 2.

Create the smallest program possible that takes an n and calculates the nth term of Sylvester's Sequence. Standard input, output and loopholes apply. Since the result grows very quickly you are not expected to take any term of which the result would cause an overflow in your chosen language.

## Test Cases

You may use either zero or one indexing. (Here I use zero indexing)

>>0
2
>>1
3
>>2
7
>>3
43
>>4
1807

• What inputs are expected to be handled? The output grows quite rapidly. – Geobits Aug 26 '16 at 1:44
• @Geobits you are expected to handle as much as your language can – Wheat Wizard Aug 26 '16 at 1:46
• Is an array which when indexed with n returns the nth number of the sequence accepted? – user6245072 Aug 26 '16 at 7:05
• @user6245072 No you must index your own arrays – Wheat Wizard Aug 26 '16 at 12:55

# GolfScript, 12 10 bytes

2 bytes thanks to Dennis.

~2\{.(*)}*


Try it online!

Uses a(n) = a(n-1) * (a(n-1)-1) + 1.

• ( is short for 1-; ) is short for 1+. – Dennis Aug 26 '16 at 6:00
• @Dennis Thanks, I must be a special kind of stupid. – Leaky Nun Aug 26 '16 at 9:59

# PHP 5.6+ (49 bytes)

$n;function f($n){return--$n?f($n)**2-f($n)+1:2;}  • I fail to see the reason of downvote: pastebin.com/QxPeUeXB . (Maybe that weird $n; that doesn't seem to do anything?) – manatwork Aug 26 '16 at 10:55
• I defines $n globaly; Without it, the code doesn't work – Florin Chis Aug 26 '16 at 11:26 • Doesn't? :o That I can't explain how I got correct results in my tests. (See the pastebin link in my previous comment. There you can see an interactive session in PHP 5.6.25.) – manatwork Aug 26 '16 at 12:05 • The downvote was cast by the Community user. It flagged the answer for length and content (only a code snippet, no text description), then marked its flag as helpful when the answer was edited. This triggers the downvote. This is essentially a bug. (CC @manatwork) By the way, the code works just fine for me without the $n;, both locally and on Ideone. – Dennis Aug 26 '16 at 17:27
• @Dennis, of course it works. In PHP a function can access global variables using the global keyword or the $GLOBALS array. – manatwork Aug 26 '16 at 17:38 # MATL, 12 bytes This can definitely be golfed further Hiq:"tpQh]0)  This solution uses 1-based indexing. Try it online! ## PowerShell v2+, 56 bytes param($n)$a=,2;0..$n|%{$a+=($a-join'*')+'+1'|iex};$a[$n]


Iterative version. Takes input, sets the first value in our array $a, loops. Each loop we take all of $a, -join them together with *, tack on a +1, and pipe to Invoke-Expression (similar to eval). That's stored as a new value on the end of $a. Then, we just index into $a for the requested number.

Calculates one index higher than necessary, which shouldn't be a problem. Output is solid until you reach the limits of round-trip precision issues and/or formatting issues where PowerShell converts to scientific notation.

PS C:\Tools\Scripts\golfing> 0..10|%{"$_ -> "+(.\sylvesters-sequence.ps1$_)}
0 -> 2
1 -> 3
2 -> 7
3 -> 43
4 -> 1807
5 -> 3263443
6 -> 10650056950807
7 -> 1.13423713055422E+26
8 -> 1.28649386832787E+52
9 -> 1.65506647324521E+104
10 -> 2.73924503086033E+208


The recursive version is one byte longer, at 57 bytes, using PowerShell's equivalent of a lambda -- $x={param($n)if(!$n){2}else{(&$x(--$n))*((&$x($n))-1)+1}}. Call it via something like &$x(4)

You could tack on a [bigint] for the iex expression to carry forward the good precision as follows -- param($n)$a=,2;0..$n|%{$a+='[bigint]"'+($a-join'"*"')+'"+"1"'|iex};$a[$n] for 73 bytes (corrected thanks to Brad Gilbert b2gills). PS C:\Tools\Scripts\golfing> 0..10|%{"$_ -> "+(.\sylvesters-sequence.ps1 $_)} 0 -> 2 1 -> 3 2 -> 7 3 -> 43 4 -> 1807 5 -> 3263443 6 -> 10650056950807 7 -> 113423713055421844361000443 8 -> 12864938683278671740537145998360961546653259485195807 9 -> 165506647324519964198468195444439180017513152706377497841851388766535868639572406808911988131737645185443 10 -> 273924503086030314234102342916746862811943643675809146279473679416086920262269936343321184045824386349295487372839923697584879743063177305807538834294603 44956410077034761330476016739454649828385541500213920807  • I'm getting different answers for f(9) and f(10) in the Rakudo implementation of Perl 6 on MoarVM f(9)==165506647324519964198468195444439180017513152706377497841851388766535868639572406808911988131737645185443 f(10)==27392450308603031423410234291674686281194364367580914627947367941608692026226993634332118404582438634929548737283992369758487974306317730580753883429460344956410077034761330476016739454649828385541500213920807 – Brad Gilbert b2gills Aug 26 '16 at 14:56 • @BradGilbertb2gills Casting issue. Need to surround the numbers in quotes so the numbers are passed to .TryParse() correctly, otherwise they're on-the-fly converted to [double] and then to [bigint] which loses precision. Thanks for the catch! – AdmBorkBork Aug 26 '16 at 15:25 # PARI/GP, 29 25 bytes a(n)=prod(i=0,n-1,a(i))+1  Thanks to alephalpha for shaving off 4 bytes. • a(n)=prod(i=0,n-1,a(i))+1 – alephalpha Aug 26 '16 at 14:54 • @alephalpha Much better! Would you like to give that as an answer, or should I edit it in? – Charles Aug 26 '16 at 15:00 • Feel free to edit it in. – alephalpha Aug 26 '16 at 15:02 # Maple 35 bytes f:=n->if(n>0,f(n-1)^2-f(n-1)+1,2)  Usage: > f:=n->if(n>0,f(n-1)^2-f(n-1)+1,2): > seq(f(i),i=0..4); 2, 3, 7, 43, 1807  ## Javascript (ES6), 25 bytes f=n=>n--?(n=f(n))*--n+1:2  Returns: • the exact result for n=0 to n=6 • an approximated value for n=7 to n=10 • Infinity for n>10 f=n=>n--?(n=f(n))*--n+1:2 console.log(f(0)) console.log(f(1)) console.log(f(2)) console.log(f(3)) console.log(f(4)) console.log(f(5)) console.log(f(6)) console.log(f(7)) # Retina, 43 26 bytes Input and output are in unary (input in 1's, output in x's.) The result is computed using a(n+1) = a(n) * (a(n) - 1) + 1, iterated n times. ^ xx {x(?=.*1)$$ x1 xx  Try it online Input and output in decimal (53 36 bytes): .*$*
^
xx
{x(?=.*1)
$$
}x1
xx
.


Try it online

Thanks to Martin for golfing 17 bytes

# Dyalog APL, 15 bytes

{×⍵:1+×/∇¨⍳⍵⋄2}


×⍵: if the argument is grater than zero:
1+ one plus
×/ the product of
∇¨ this function applied to each of
⍳⍵ first n integers (beginning with zero)
⋄ else:
2 return two

0-based indexing – needs ⎕IO←0.

TryAPL online!

• I had tried {1+×/∇¨⍳⍵} before, but that resulted in a WS FULL message. Why does iterating over an empty list cause infinite recursion though? It works as (I) expected with ngn-apl. – Dennis Aug 27 '16 at 22:45
• @Dennis It is because empty lists are not really that empty: They carry information about their prototype element, obtainable with ⊃EmptyList. Thus, for user-defined functions, the function is called once on empty lists, to determine the prototype. Interestingly, this leads to the possibility of storing any amount of arbitrary information in an "empty" variable. – Adám Aug 27 '16 at 22:50
• – Adám Aug 27 '16 at 23:12

## Pip, 11 bytes

Lao*:o+1o+1


The shortest way I've found so far. Uses the formula from Martin's Hexagony answer: define b(0) = 1, b(n) = b(n-1) * (b(n-1) + 1), and then a(n) = b(n) + 1.

La           Loop number of times equal to cmdline input:
o*:o+1     Multiply o by o+1 in place (o is a variable preinitialized to 1)
o+1  Output o+1


Try it online!

# Perl, 282623 22 bytes

Includes +1 for -p

Run with the input (with 1-based indexing) on STDIN:

sylvester.pl <<< 5


sylvester.pl:

#!/usr/bin/perl -p
$.*=$_=$.+1for($_)x$_  ## R, 56 49 bytes n=scan();x=2;for(i in 1:n){x=c(x,prod(x)+1)};cat(x[n+1])  ### Ungolfed: n=scan() # Take input n x=2 # Initialize sequence to 2 for(i in 1:n){ x=c(x,prod(x)+1) # Append the product of the previous numbers + 1 } cat(x[n+1]) # Print the nth + 1 number in seq  Slightly golfed thanks to @Frédéric: n=scan();x=2;for(i in 1:n)x=c(x,prod(x)+1);x[n+1]  • You could golf out some bytes by removing the {...} of the for loop that you don't need here since there's only one instruction in. Moreover, I don't think you absolutely need the cat(...) at the end, even if it looks prettier. – Frédéric Aug 26 '16 at 13:52 # Python 2, 87736460 51 bytes 14 23 25 bytes saved thanks to Leaky Nun 9 bytes saved thanks to Specter Terrasbane Here's my go at my own challenge. f=lambda x:reduce(int.__mul__,[1]+map(f,range(x)))+1  • instead of using o.mul, you can use int.__mul__ – Leaky Nun Aug 26 '16 at 1:55 • You can also do it recursively. – Leaky Nun Aug 26 '16 at 1:57 • lambda x:reduce(lambda y,z:y+[reduce(int.__mul__,y)+1],[2])[-1] I don't know what you pre-initialized the array for. – Leaky Nun Aug 26 '16 at 2:12 • Also range(x-1) is equivalent to range(0,x-1) – Leaky Nun Aug 26 '16 at 2:13 • You should specify this as Python 2 due to the reduce, which would require importing functools in 3. Edit: Also it does not seem to work – Jonathan Allan Aug 26 '16 at 10:54 # Python 3, 71 69 68 63 57 bytes Python 3, 71 69 68 bytes l=[2] for _ in range(int(input())): n=1 for i in l:n*=i l+=[n+1] print(l[-1])  Also 68 bytes: l=[2] a=int(input()) while len(l)<a: n=1 for i in l:n*=i l+=[n+1] print(l[-1])  EDIT: Thanks @WheatWizard for pointing out about using n instead of no, and removing the space between for i in l and n*=i. Also thanks for pointing out about moving int(input()) into the range function. EDIT 2: Thanks @WheatWizard for pointing out the iteration tip, it has allowed me to write these two, shorter, programs: Python 3, 63 bytes l=[2] for _ in"a"*int(input()): n=1 for i in l:n*=i l+=[n+1] print(l[-2])  # Python 3, 57 bytes l=[2] for _ in"a"*int(input()): b=l[-1] l+=[(b-1)*b+1] print(b)  The second code (57 bytes) does not follow the instructions for making the sequence (i.e: product of the sequence+1) instead it works on the fact that the last number will always be the product of the rest+1 meaning that instead of iterating through the sequence, I can multiply the last number by itself-1 and then add 1 back on. • You can lose the space between the : and the no*=i. – Wheat Wizard Aug 28 '16 at 14:47 • plus no can be renamed to n – Wheat Wizard Aug 28 '16 at 14:47 • You could also move the int(input()) into the range() and print l[-1] instead of l[a] – Wheat Wizard Aug 28 '16 at 14:50 • You seem to have forgotten to make changes on line 5. – Wheat Wizard Aug 28 '16 at 14:55 • The space is still there... – Wheat Wizard Aug 28 '16 at 15:08 # dc, 34 bytes 0sg[[d1-*1+Lgx]Sg1-d0<f]dsfx2Lgx++  This pops the argument from top of stack, and pushes the result to stack, as normal for dc. ### Annotated full program #!/usr/bin/dc # accept input ? # initialise the bottom of the g-stack 0sg # Add n iterations of recurrence formula [[d1-*1+Lgx]Sg1-d0<f]dsfx # Prime the 0th value 2 # Execute all of the g-stack Lgx # Last instruction left a zero on the stack + # Special case: if input is 0, f left a -1 behind. This is a correction # for wrongly doing 2->3 in that case + # print output p  It works by using the recurrence relation described in OEIS: a[n+1] = a[n]² - a[n] + 1, starting with a[0]==2. Equivalently, a[n+1] = a[n](a[n]-1) + 1, written in dc as d1-*1+. We push n copies of the program d1-*1+ to the stack g, prime the main stack with the initial value 2, and set off. There's a correction for n=0, because we always push at least one instance of the recurrence. Handily, we can fix that, because function f leaves -1 on the stack in that case, and 0 otherwise. ### Test output: 0: 2 1: 3 2: 7 3: 43 4: 1807 5: 3263443 6: 10650056950807 7: 113423713055421844361000443 8: 12864938683278671740537145998360961546653259485195807 9: 165506647324519964198468195444439180017513152706377497841851388766535868639572406808911988131737645185443 10: 27392450308603031423410234291674686281194364367580914627947367941608692026226993634332118404582438634929548737283992369758487974306317730580753883429460344956410077034761330476016739454649828385541500213920807 11: 750346333909286311464218348364293017384724140073732363176684391768374238237200233203724274839819736227493060107386942069521875902258281351952761393460726027774387698896086030486687796275661950199835484418384103096899499524666007073298797852932127876923983340497448231960048833094195425231846478785035602339261149953564729371337917773386670133413581537490788020231265093210310224397095644371148893261284201611453610443 12: 563019620811106188735345793029131127645126456201508328395934709948713369875017428398616526515350801978901280848429236023780360625686785326318172135941491985234945009397938528773976977887356630327917721419666751591559883455828027643329740930320548089836575156581070880066847294722353255119648341271402723167206816552066010592134725124376810874159292118224274440907862996344067325025151751506647339684959479438354192989470470488621172745980086435801519575581664038703641801974453354864238123148678312993828328158261237284571445163949954735321181426755485563860804935755099928085508785637606452207007984638610604113718738727804169240213041488645142155664706600019329809186121988322464207409749450811638629159492106903853533242008723387118397794980575878357156285111258733862431522178328441469980110808406224908967784943255608168545045807 13: 316991093418281796980738587324498503465985663563441351614965595710659626509023026378810664917769979749230375616949095839488230630774365784766585678394020150307543448990686716629735821013846949520218610883960194258122308878890968113466913053158083723070515602776324450012006168592325685889943262781933132780014500164781142497008320197365141855119010118077705678576368755229432787283587491897170937657553086679827144799087397199129432233281774986701578182361474362882421927116451702868297347253630717771757957438615228337552613006602138504913204731416596016919892552584014059657038635241014722034375088903021315484876024289828564177721518967548558336213757430766087730295001595138792039193609746512786881659903942227205537012891959828532940594409566163694454653037016800027294499421291400492041597627904670868358757293568071192353450861046220903121264904877170553117406388566305811088048722332066039563082830094354238880920553702156022717609505663704069061610519598560624334935881397512984533454362321248854729174814176943381209734396537109715522733592519084452225492188715705481658124794599731527140827530319488791822858978924780015280035310521766457660381662363459675166288099343838361130665940967928850877954452552005403436587169730515877297387774041696625714088910160966686976352250237188765370786055553236020704945964404004577531357576565107610684603955726197582253686411381871520005412537060114403951740688011605681233186481569161991920354077827812176666950510226500521720520469387009627382085749910481677989034236154462513159644551504513525508614268492165297534777956118363272320828495465015734808069866823697938666550506233479919235237378935603448169915183235443 14: 100483353306517857188816511171973264968130481955993490961776578914384855009465596512878627978186320098524674810822809215084298821790532480217571435837318997038501027166944291436298311859998927368878478765662052261013905379683606908563327049366289350227874801320137528476359434996625155321924896471487253609774383161294073700386567438895729619959183688827019175395384990049660760826706275656247200320644562229907983316938796200727194177710641698165578469600709422183645673081099011178658794923207340118316226811044120124994280172667435132674250477913878334229469634774955766654959747171269329356217729819952093850230832136940392143299736102334976183356619572749168857678782222255344301700168226608019630704240193738779198063738629831841413772846575231497629810592678428725195220732911880790353530776313807267315574488856933933931968726196972550157616749777588888821303729634383570842953721086839502335017753789603211887639264756326943906234925564059713122739441052488376985825209930025027331642643833862623818612641648534007690784009230583997364206027891426162271405399576520661523570494921380174573570295575834537952976231451138917903119690734728050868143651943079046629073247009723903287946523072583428647088990713577505274669321894108556505593299658881909461644436220683735665670039545950199079805499092674406353548516096407916275052371510047476698658893306349068133879014809524920709542592832744670134019181012820484031829100985336532444887396195766078330294774788615792775829183741076029977456631131003360810557200089618493958319779112283680033864350359591347851857639767660909655544622503481811302270128415655545885368248621757558781407774929247336942808205264958129961178967782261436710020914320953492879039247990865098954170193648898076402861312803632817456574773061811978648089614340109760702469211985641224691799075765885167117855362986656144090527289697873066081524676142089795828969028242126025160985084779781007974719610529121366009147473928798323429514961662955075830495059188425232901129844306717176754536658533793214326255403412914373598713551532524170357661947859352181097102731800860434424406585016086605044703266455861041380556886689319004123095655013285849524741495205262970148746426581276319938997140587845485798731493239638046736464142096404810367841352036464369335995230683471276720095626585363049720580305501778636182605592068492151793190258602236893550140479021220581841241476297575889705144565984983772835663470728331437295945770147420128139970898197153180098987682528952187399111614545313650329728193759220800741639763444502297436225484681500013482802572032911312979385547883726365017125898034265710900263014307543995711388447583272341504352098242603013317289071851486202200363353430637400418445403028209289752937101610106546407815385279051077386957963745649023423516876644029840971378691516442745761797057396994687382618710236478541855961623090443801434726418529440871490091117324165342131670988126442133481305788646667033011070351616554711007362680081909808994105138595194669575127185645557552945017168697977343707144183186705714691228418618276386168995938283316172058534438295911377445943568762495111242585896098376432273844229629284777619272064792008892485171141118486114024760197688360378085790695965894817048250719620713759873671293320363983916413298621510321025850259140267373664767105414000002388170807 15: 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16: (13341 digits) 17: (26681 digits) 18: (53361 digits) 19: (106721 digits) 20: (213441 digits) 21: (426881 digits) 22: (853761 digits) 23: (1707522 digits) 24: (3415043 digits) 25: (6830085 digits) 26: (13660169 digits) 27: (27320338 digits) 28: (54640675 digits) 29: (109281349 digits)  As you can see, these numbers grow quite rapidly; the computation time also increases in the same proportion as the result length. It took me half a minute to compute a(23), and several hours for a(29). # JavaScript (ES2016), 25 bytes f=x=>x--?f(x)**2-f(x)+1:2  Uses the zero-indexed sequence. Here's an example: f=x=>x--?f(x)**2-f(x)+1:2 const output = document.getElementById('output'); for(let i = 0; i < 10; i++) output.textContent += 'f(' + i + ') = ' + f(i) + '\n'; <pre id="output"></pre> # Groovy, 26 bytes This answer was basically stolen from @Geobits. a={n->n--?a(n)*~-a(n)+1:2}  Took that answer and, using groovy shortcuts, optimized it further. # Haskell, 27 bytes f 0=2 f n=f(n-1)^2-f(n-1)+1  Try it online! Not the shortest Haskell answer but it uses a new method. # Common Lisp, 53 bytes (defun f(n)(do((x 2(1+(* x(1- x)))))((<(decf n)0)x)))  The result is computed through the formula a(n+1) = a(n) * (a(n) - 1) + 1, starting from 2 and iterated n times. # K (oK), 12 bytes {2|1+*/o'!x}  Try it online! Blatantly stolen from Port of Adám's answer. Prints up to the first 11 elements of the sequence, after which the numbers become too big to handle. Thanks @ngn for 6 bytes! ### How: {2|1+*/o'!x} # Main function, argument x. o'!x # recursively do (o) for each (') of [0..x-1] (!x) 1+*/ # 1 + the product of the list 2| # then return the maximum between that product and 2.  # Ruby, 29 bytes f=->a{a<1?2:(b=f[a-1])*~-b+1}  +2 for adding f= to byte count. -2 from Dingus's suggestion. Try it online! • Nice, but you need to include f= in the byte count since f is called recursively. You can completely offset the 2 bytes gained from f= by replacing ==0 with <1 and storing the value of f[a-1] (i.e. (b=f[a-1])*~-b). – Dingus Aug 12 at 6:24 ## Perl6, 49 bytes my &f={my @c=2;@c.push: 1+[*] @c for ^$_;@c[*-1]}


Ungolfed:

sub f($n) { my @sylvester = 2; # declare an array to store the previous # values of the sequence # insert S(1) ... S(n) (n iterations) @sylvester.push(1 + [*] @sylvester) for 0..^$n;
return @sylvester[* - 1]; # return last element
}

• You don't need to include my&f= in these competitions, an anonymous code object is fine. – Brad Gilbert b2gills Aug 26 '16 at 22:13

## Batch, 70 bytes

@set/ap=r=2
@for /l %%i in (1,1,%1)do @set/at=p,p*=r,r=t+1
@echo %r%


Zero indexed, uses @LeakyNun's recurrence relation. Conveniently set/a works inside a for loop, because I'm not using % substitution.

# Scala, 85 bytes (51?)

val s:Stream[BigInt]=2#::3#::s.tail.map{c=>c*c-c+1}
print(s.take(args(0).toInt).last)


3 less if I stick with an Int. The stream bit which actually does all the work is 51 bytes. The rest is just printing

Usage:

$scala sylvester.scala 20  # Javascript (Using external library - Enumerable) (52 bytes) n=>_.Sequence(n,(i,a)=>_.From(a).Product()+1).Last()  Link to lib: https://github.com/mvegh1/Enumerable Code explanation: Static method on library "Sequence" will generate a sequence of elements for a count of 'n', according to the predicate accepting params "i"teration, "a"ccumulated-array. The predicate states to cast the array to the library's Enumerable data-type, use the built in Product method on it, then add 1 to that value. After the sequence is generated, take the last value because the problem states to just find the 'N'th element of the sequence # Sesos, 17 bytes Hexdump: 0000000: ae2cf0 20f8be b273d0 7d9cde a0dd3b 7e3e .,. ...s.}....;~>  Try it online! # UGL, 15 bytes cuuild@$d*u@:_o


Try it online!

## Pyke, 9 bytes

[SOm[BhRK


Try it here!

## Perl, 35 bytes

$a=2;$a=$a**2-$a+1 for 1..<>;say \$a


Might be shorter with a recursive function but I couldn't get that to work in my few attempts

# ForceLang, 193 bytes

Noncompeting, as for some reason I forgot to give the language's lists a len method when I first implemented them.

def s set
def l a.len()
s g goto
s n n+1
s a []
s b 1
label 1
s n n+-1
s a[l] b+1
if n=0
g 3
s b 1
s i 0
label 2
s b b.mult a[i]
s i i+1
if i=l
g 1
g 2
label 3
io.write a[l+-1]
`