Labyrinth, 18 bytes

Credits to Sp3000 who found the same solution independently.

? # )}"{!@ * ( (:{ 

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Perl, 28 26 23 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$_ 

Python 2, 87 73 64 60 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 

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.

Flurry, 26 bytes

{}[(<><<>()>)({})]{{}}{}{} 

Verification

$ echo -n "{}[(<><<>()>)({})]{{}}{}{}" | wc -c 26 $ ./flurry -nin -c "{}[(<><<>()>)({})]{{}}{}{}" 0 2 $ ./flurry -nin -c "{}[(<><<>()>)({})]{{}}{}{}" 1 3 $ ./flurry -nin -c "{}[(<><<>()>)({})]{{}}{}{}" 2 7 $ ./flurry -nin -c "{}[(<><<>()>)({})]{{}}{}{}" 3 43 $ ./flurry -nin -c "{}[(<><<>()>)({})]{{}}{}{}" 4 1807 

How it works

We have a formula \$a(n+1) = a(n) (a(n) - 1) + 1\$, but pred is well-known to be expensive in SKI calculus, so I defined an auxiliary sequence:

$$ b(n) = a(n)-1 \\ a(n+1) - 1 = (a(n)-1+1)(a(n)-1) \\ b(n+1) = b(n)(b(n)+1), b(0) = 1 $$

Then we can use the nature of Church numerals to calculate the \$n\$-th term as

b = \n. n (\x. x ∘ succ x) 1 a = \n. succ (b n) = \n. succ (n (\x. x ∘ succ x) 1) 

where succ is the successor function and ∘ is function composition (which acts as multiplication when given two Church numerals).

We can represent the \x. x ∘ succ x part directly in Flurry as {<({})[<><<>()>{}]>}, but we can do better. We can try converting to SKB:

\x. \f. x (succ x f) \x. \f. K x f (succ x f) \x. S (K x) (succ x) \x. (S∘K) x (succ x) S (S∘K) succ succ succ (because succ = S (S∘K)) 

So we can duplicate succ via the stack, making it [(<><<>()>){}] (-6 bytes). Using it in the full program, we get:

\n. succ (n (\x. x ∘ succ x) 1) \n. succ (n (succ succ) 1) <><<>()>[ succ ( {}[ n ( (<><<>()>) succ [push succ] {} succ [pop] ] ) {} I [pop from empty stack] ] ) 

But I wasn't satisfied because it still has two copies of succ. So I changed the "calculate b(n) and apply succ on it" to "apply succ b(n) times to 1", which gives

\n. n (succ succ) 1 succ 1 {}[ n ( (<><<>()>) succ [push succ] ({}) succ [pop and push succ] ] ) {{}} I [1] {} succ [pop] {} I [pop from empty stack] 

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.

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Husk, ¹¹ 7 bytes

!¡o→Π;2 

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1-indexed.

-4 bytes from Leo.

Vyxal, 7 bytes

-2 bytes thanks to Steffan

2$(:‹*› 

Explanation:

2$(:‹*› 2$ Push 2 and swap it with the input (: Loop through input and duplicate value ‹* Decrement the value and multiply › Increment value 

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Julia 1.0, 27 22 bytes

~x=*(1,.~(0:x-1)...)+1 

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-5 bytes (!!) thanks to MarcMush: Replace prod() with *(); also, pad the argument list with 1 to avoid handling \$a_0\$ as a special case

Actually, 14 12 bytes

This used 0-indexing. Golfing suggestions welcome. Try it online!

2#,`;πu@o`nF 

Ungolfing:

2# Start with [2] ,` `n Take 0-indexed input and run function (input) times ; Duplicate list πu Take product of list and increment @o Swap and append result to the beginning of the list F Return the first item of the resulting list 

GolfScript, 12 10 bytes

2 bytes thanks to Dennis.

~2\{.(*)}* 

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

MATL, 12 bytes

This can definitely be golfed further

Hiq:"tpQh]0) 

This solution uses 1-based indexing.

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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 

PARI/GP, 29 25 bytes

a(n)=prod(i=0,n-1,a(i))+1 

Thanks to alephalpha for shaving off 4 bytes.

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 

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Input and output in decimal (53 36 bytes):

.* $* ^ xx {`x(?=.*1) $`$` }`x1 xx . 

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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.

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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 

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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] 

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.

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.

AWK, 32 bytes

{for(n=2;$0>i++;n=n*n-n+1);}$1=n 

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Vyxal m, 5 bytes

λvxΠ› 

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Jelly port.

Thunno 2, 7 bytes

2${D⁻×⁺ 

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Port of Emigna's 05AB1E answer.

Explanation

2${D⁻×⁺ # Implicit input 2 # Start with 2 ${ # Input number of times: D⁻× # Multiply by x-1 ⁺ # And increment # Implicit output 

PHP 5.6+ (46 bytes)

function f($n){return--$n?f($n)**2-f($n)+1:2;} 

Japt -g, 6 bytes

Takes input as a singleton integer array, 1-indexed.

ÇÆiU×Ä 

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Japt -h, 6 bytes

Takes input as an integer, 1-indexed.

ÆßX ×Ä 

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Original w/o flag, 9 8 bytes

Takes input as an integer, 0-indexed.

@ÒZ×}gNÅ 

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Nekomata, 7 bytes

2ᶦ{:←*→ 

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Takes no input, and outputs all terms of the Sylvester's sequence.

2ᶦ{:←*→ 2 Start with 2 ᶦ{ Infinite loop: : Duplicate ← Decrement * Multiply → Increment