To exponential digit growth and beyond!

Challenge

Given a base $$\1 < b < 10\$$ and an index $$\t \ge 1\$$, output term $$\x_t\$$, defined as follows:

• $$\x_1 = 11_{10}\$$
• $$\x_{i+1}\$$ is obtained by converting $$\x_i\$$ to base $$\b\$$ and then reinterpreting its digits in base $$\10\$$
• Output should be in base $$\10\$$

A walk through for base 5, term 5 would be:

• $$\x_1 = 11_{10}\$$.
• $$\11_{10} = 21_{5}\$$ so $$\x_2 = 21_{10}\$$.
• $$\21_{10} = 41_{5}\$$ so $$\x_3 = 41_{10}\$$.
• $$\41_{10} = 131_{5}\$$ so $$\x_4 = 131_{10}\$$.
• $$\131_{10} = 1011_{5}\$$ so $$\x_5 = 1011_{10}\$$.
• We output the string "1011" or the integer 1011.

Test Cases

Note: these are one indexed

base 2, term 5 --> 1100100111110011010011100010101000011000101001000100011011011010001111011100010000001000010011100011
base 9, term 70 --> 1202167480887
base 8, term 30 --> 4752456545
base 4, term 13 --> 2123103032103331200023103133211223233322200311320011300320320100312133201303003031113021311200322222332322220300332231220022313031200030333132302313012110123012123010113230200132021023101313232010013102221103203031121232122020233303303303211132313213012222331020133


Notes

• Standard loopholes are not allowed
• Any default I/O method is allowed
• You may use different indexes (such as 0-indexed, 1-indexed, 2-indexed, etc) for $$\t\$$
• You may output the first $$\t\$$ terms.
• As this is , shortest code wins for that language
• Do we have to support larger numbers or just numbers up to 2^31 - 1? – Gymhgy May 19 '19 at 23:39
• @EmbodimentofIgnorance The maximum of your language (Remember the standard loophole, though!) – MilkyWay90 May 19 '19 at 23:43
• Is there a challenge that includes bases > 10? (In that case you would repeatedly interpret 11 as if it was in base b and convert it back to base 10, etc.) – Neil May 21 '19 at 11:11
• @Neil I didn't include bases higher than 10 since (for example) 4a wouldn't be a valid number in base-10 – MilkyWay90 May 22 '19 at 22:49
• You wouldn't get 4a, since you'd be interpreting the base 10 digits as base b and converting to base 10 each time (i.e. the other way around from this question). – Neil May 22 '19 at 22:59

JavaScript (Node.js), 40 bytes

Thanks to @Neil for saving 5 bytes on this version and 2 bytes on the BigInt version

Takes input as (t)(base), where $$\t\$$ is 1-indexed.

n=>g=(b,x=11)=>--n?g(b,+x.toString(b)):x


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JavaScript (Node.js), 48 bytes (BigInt version)

Takes input as (t)(base), where $$\t\$$ is 1-indexed. Returns a BigInt.

n=>g=(b,x=11n)=>--n?g(b,BigInt(x.toString(b))):x


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• Do you need to eval in the first version? + would save 5 bytes... – Neil May 20 '19 at 0:36
• And BigInt saves two bytes in the second version, because you don't need to add n to the string. – Neil May 20 '19 at 0:37
• (b,t,x=11)=>--t?f(b,t,+x.toString(b)):x is 1 char shorter – Qwertiy May 25 '19 at 21:45
• @Qwertiy It's actually 1 byte longer, because we'd need to prepend f= (since the function is referencing itself). – Arnauld May 25 '19 at 22:04
• @Arnauld, oops. Then this one n=>b=>g=(x=11n)=>--n?g(BigInt(x.toString(b))):x :) If call f(t)(b)() is allowed. – Qwertiy May 25 '19 at 22:07

05AB1E, 5 bytes

>IF¹B


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Explanation

>       # increment <base>
IF     # <term> times do:
¹B   # convert from base-10 to base-<base>


Note that there is no need to explicitly start the sequence at 11.
Starting at base+1 and doing an extra iteration will result in the first iteration giving 11.

Japt, 9 bytes

ÆB=sV n
B


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(Two inputs, U and V)
Æ            Range [0..U)
B=          For each, set B (B is preinitialized to 11) to
sV          B's previous value converted to base-V
n           and back to base-10
B            Print out B's final value

• This will never be able to output the first term, will it? – Shaggy May 20 '19 at 7:14
• @Shaggy Fixed at the cost of two bytes – Gymhgy May 20 '19 at 21:57
• Nicely saved :) Wouldn't have thought of doing that meself. – Shaggy May 21 '19 at 22:25

Husk, 7 bytes

!¡odB¹→


Try it online! This version is 2-indexed; it takes the two inputs $$\b\$$ and $$\t\$$ as arguments.

Explanation

!¡odB²→²⁰  (Written with explicit arguments; let B and T denote them.)
!       ⁰  Take element T of
¡          the function repeatedly applied to
→²     B + 1:
o B²       Convert to base B
od         then convert to decimal.


Wolfram Language (Mathematica), 46 bytes

bNest[FromDigits[#~IntegerDigits~b]&,11,#]&


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Call with f[base][t]. 0-indexed.

Retina, 67 bytes

.+,(\d+)
11,$1* "$+"{^\d+
*
)+(?=_.*,(_+))(\1)*(_*)
$#2*_$.3
,_+



Try it online! Takes comma-separated inputs $$\t\$$ (0-indexed) and $$\b\$$. Does all of its calculations in unary so times out for large numbers. Explanation:

.+,(\d+)
11,$1*  Initialise $$\x_0=11\$$ and convert $$\b\$$ to unary. "$+"{


Repeat $$\t\$$ times.

^\d+
*


Convert $$\x_i\$$ to unary.

)+(?=_.*,(_+))(\1)*(_*)
$#2*_$.3


Convert to base $$\b\$$.

,_+



Delete $$\b\$$ from the output.

Python 2, 71 bytes

def f(b,n):h=lambda x:x and x%b+10*h(x/b);return n and h(f(b,n-1))or 11


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0-indexed.

Clojure, 109 bytes

Credit to MilkyWay90 for removing 10 bytes by spotting unnecessary spaces Credit to Embodiment of Ignorance for another byte from another unnecessary space

Golfed

(defn f([b t](f b t 11))([b t v](if(= t 1)v(f b(dec t)(read-string(.(new BigInteger(str v))(toString b)))))))


Ungolfed

(defn f
([base term] (f base term 11))
([base term value] (if (= term 1)
value
(f base (dec term) (read-string (. (new BigInteger (str value)) (toString base)))))))


I think the main place bytes could be saved is the expression for... reradixing? whatever that would be called. Specifically:

(read-string (. (new BigInteger (str value)) (toString base)))

• Do you need those spaces? Can you eliminate spaces? – MilkyWay90 May 20 '19 at 14:32
• Didn't even occur to me to remove spaces where they were separating syntactically differentiable things; looks like Clojure is a bit more permissive than I thought. Thank you! – user70585 May 20 '19 at 17:50
• There is still an unnecessary space in (if (= t 1) – Gymhgy May 20 '19 at 21:59
• Whoop, good catch 👍 – user70585 May 21 '19 at 18:51
• 93 bytes – Gymhgy May 22 '19 at 3:01

Perl 6, 28 bytes

{(11,+*.base($^b)...*)[$^t]}


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The index into the sequence is zero-based.

Jelly, 8 7 bytes

‘       | b + 1
$⁴¡ | Repeat the following t times b³ | Convert to base b Ḍ | Convert back from decimal to integer  • Explanation for those of us who can't recognize atoms at a quick glance? – MilkyWay90 May 19 '19 at 23:43 K (ngn/k), 13 bytes {y(10/x\)/11}  Try it online! Pyth, 8 bytes uijGQTEh  Try it online! C# (Visual C# Interactive Compiler), 87 bytes n=>m=>{int g=11;for(var s="";m-->0;g=int.Parse(s),s="")for(;g>0;g/=n)s=g%n+s;return g;}  Saved 5 bytes thanks to @KevinCruijssen Try it online! • 87 bytes by changing the do-while into a regular for-loop. – Kevin Cruijssen May 20 '19 at 9:52 brainfuck, 270 bytes ++<<++<,+++<-[----->-<]<,,[<-----[->++++++++++<]++>[-<+>],]<[>>>>>>[<<<[->>+<<]>>>>>]<<[[>+<-]>>[-[<++++++++++>-]>+>]<[<<]>>[-<<+>>>+<]>>[-[<-[>>+>>]>>[+[-<<+>>]>[-<]<[>]>++>>>]<<<<<-]+>[-<+<+>>]<<[->>+<<]>>>]<[-]<[[-<+>]<<]<]<[->>+<<]<-]>>>>[>>]<<[>-[-----<+>]<----.<<]  Try it online! 0-indexed. The number of iterations is assumed to be at most 255. Explanation The tape is laid out as follows: num_iterations 0 0 base digit0 0 digit1 0 digit2 ...  Each digit is actually stored as that digit plus 1, with 0 reserved for "no more digits". During the base conversion, the digits currently being worked on are moved one cell to the right, and the base is moved to the left of the current working area. ++<<++ Initialize initial value 11 <,+++<-[----->-<] Get single digit as base and subtract 48 to get actual number <,,[<-----[->++++++++++<]++>[-<+>],] Read multiple digits as number of iterations < Go to cell containing number of iterations [ For each iteration: >>>>>> Go to tens digit cell [<<<[->>+<<]>>>>>] Move base to just before most significant digit << Return to most significant digit [ For each digit in number starting at the left (right on tape): [>+<-] Move digit one cell to right (to tell where current digit is later) >>[-[<++++++++++>-]>+>] Multiply each other digit by 10 and move left <[<<]>> Return to base [-<<+>>>+<] Copy base to just before digit (again) and just before next digit to right (left on tape) >>[ For each digit at least as significant as this digit: -[<-[>>+>>]>>[+[-<<+>>] Compute "digit" divmod base >[-<]<[>]>++ While computing this: add quotient to next digit; initialize digit to "1" (0) first if "0" (null) >>>]<<<<<-] End of divmod routine +>[-<+<+>>] Leave modulo as current digit and restore base <<[->>+<<] Move base to next position >>> ] <[-]< Delete (now useless) copy of base [[-<+>]<<]< Move digits back to original cells ] Repeat entire routine for each digit <[->>+<<] Move base to original position <- Decrement iteration count ] >>>>[>>]<<[>-[-----<+>]<----.<<] Output by adding 47 to each cell containing a digit  Charcoal, 14 bytes ≔11ζＦＮ≔⍘ＩζＩηζζ  Try it online! Link is to verbose version of code. Takes inputs as $$\t\$$ (0-indexed) and $$\b\$$. Explanation: ≔11ζ  $$\x_0=11\$$. ＦＮ  Loop $$\b\$$ times. ≔⍘ＩζＩηζ  Calculate $$\x_i\$$. ζ  Output $$\x_t\$$. Pari/GP, 50 bytes (b,n)->x=11;for(i=2,n,x=fromdigits(digits(x,b)));x  Try it online! C (gcc), 59 bytes f(t,b){t=t?c(f(t-1,b),b):11;}c(n,b){n=n?c(n/b,b)*10+n%b:0;}  Try it online! Groovy, 45 bytes a,b,c=11->a--?f(a,b,a.toString(c as int,b)):c  Try it online! Port of @Arnauld's answer PHP, 83 75 bytes function c($b,$t,$v=11){return $t==1?$v:c($b,$t-1,base_convert($v,10,$b));}


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This one will only work work with "small" numbers (e.g. not test cases 1 and 4)

Japt, 10 bytes

0-indexed. Takes t as the first input, b as the second.

_ìV ì}gBìC


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Gaia, 8 bytes

Bd
11@↑ₓ


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Takes 0-based iterations then base.

Bd		| helper function: convert to Base b (implicit) then convert to base 10
| main function:
11		| push 11
@		| push # of iterations
↑ₓ		| do the above function (helper function) that many times as a monad


Ruby, 39 bytes

Zero-indexed.

->b,n,x=11{n.times{x=x.to_s(b).to_i};x}


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Perl 5-Mbigint -pa, 65 bytes

$\=11;map{$p=$\;$\%=0+"@F";$\=($p%"@F").$\while$p/=0+"@F"}2..<>}{


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GolfScript, 16 bytes

11\~{base""*~}+*


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The input is t (0-indexed) and then b.

11                 # x_1
\~               # Parse the inputs to integers
{base""*~}+    # Add b to the beginning of this block to get {b base""*~}
{b base""*~}     # This block goes from x_n to x_(n+1)
b base          # Convert to base b
""*       # Join the digits with "" between them
~      # Parse it to an integer
*   # Execute this block t times