# Start from ones

Given a strictly positive integer n, follow these steps:

1. Create an array A with n 1s.
2. If A only has one element, terminate. Otherwise, starting from the first element, replace each pair of A with its sum, leaving the last element as is if A's length is odd, and repeat this step.

The output should contain A's state after each step in order from the first step to the last. Usage of standard loopholes is forbidden. This is a challenge, so the solution with the fewest bytes in each language wins.

## Test cases

Each line in the output of these examples is a state. You can output via any reasonable format.

Input: 1

[1]


Input: 4

[1, 1, 1, 1]
[2, 2]
[4]


Input: 13

[1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1]
[2, 2, 2, 2, 2, 2, 1]
[4, 4, 4, 1]
[8, 5]
[13]


Input: 15

[1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1]
[2, 2, 2, 2, 2, 2, 2, 1]
[4, 4, 4, 3]
[8, 7]
[15]

• Can I copy this questions idea for the reverse order? Given number n, output stepwise A, and so on until you reach n 1s? – pixma140 Aug 22 at 11:22
• @pixma140 That would be essentially the same challenge, just with the output reversed afterwards. The modification is trivial. – Erik the Outgolfer Aug 22 at 11:39

# 05AB1E, 7 bytes

Å1Δ=2ôO


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# MATL, 10 bytes

:gt2estnq


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### How it works

:     % Input n (implicit). Range [1 2 ... n]
g     % Convert to logical. Gives [1 1 ... 1]
% Do...while
t   %   Duplicate
2   %   Push 2
e   %   Reshape as 2-column matrix, in column-major order, padding with 0 if needed
s   %   Sum of each column
t   %   Duplicate
n   %   Number of elements
q   %   Subtract 1. This will be used as loop condition
% End (implicit). If top of the stack is not zero run new iteration
% Display stack, bottom to top (implicit)


# Python 3, 57 bytes

def f(i,j=1):print(i//j*[j]+[i%j][:i%j]);i>j and f(i,j*2)


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# Python 2, 51 bytes

def f(i,j=1):print i/j*[j]+[i%j][:i%j];i>j>f(i,j*2)


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-6 bytes total thanks to tsh

Recursive function. For each step, it constructs a list of powers of 2, such that the sum is smaller than or equal to the given integer. It then appends the remainder, if it is larger than 0.

• Python 3 61 bytes: def f(i,j=1):l=i//j*[j]+[i%j][:i%j];print(l);i>j and f(i,j*2); Python 2 55 bytes: def f(i,j=1):l=i/j*[j]+[i%j][:i%j];print l;i>j>f(i,j*2) – tsh Aug 23 at 4:09
• @tsh Of course, thanks! i>j didn't work in my previous solution and I forgot to try it afterwards. – Jitse Aug 23 at 6:59

# Jelly, 6 bytes

L€+2/Ƭ


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# R, 65 bytes

-1 byte thanks to Giuseppe.

n=scan();while(T<2*n){cat(rep(+T,n%/%T),if(n%%T)n%%T,"\n");T=2*T}


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Avoids recursion. In R, %/% is integer division and %% is the modulo. For each power of 2 k=2^i, we need to print n%/%k times the value k, and then n%%k if that value is non zero. Do this for all powers of 2 smaller than $$\2n-1\$$.

Here I am using T instead of k, since it is initialized as TRUE which is converted to 1. I still need to print +T instead of T to avoid a vector of TRUEs in the output.

• Beat me by about 5 minutes and almost 60 bytes... But Giuseppe is right, it doesn't output the final step. – Sumner18 Aug 22 at 13:50
• @Sumner18 Should be fixed now. – Robin Ryder Aug 22 at 14:14
• +T is shorter than T+0 – Giuseppe Aug 22 at 14:15
• @Giuseppe Thanks, I knew I was forgetting something. – Robin Ryder Aug 22 at 14:16

# Pyth, 10 bytes

.u+McN2m1


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.u          # Apply until a result is repeated, return all intermediate steps: lambda N,Y:
+M        # map by + (reduce list on +):
cN2     # chop N (current value) into chunks of 2, last one is shorter if needed
m1Q  # map(1, range(Q)) (implicit Q = input)


-1 byte thanks to FryAmTheEggman

# Wolfram Language (Mathematica), 55 54 bytes

Last@Reap[1~Table~#//.a_:>Tr/@Sow@a~Partition~UpTo@2]&


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Finally, Sow/Reap beats an alternative!

Returns a singleton list containing a list of the steps.

# K (oK), 15 17 bytes

{{+/'0N 2#x}\x#1}


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# J, 20 bytes

+/\~&_2&.>^:a:@<@#&1


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-2 bytes thanks to Bubbler

• 20 bytes by eliminating [: and (). – Bubbler Nov 14 at 9:37
• Nice catch! Thanks. – Jonah Nov 14 at 12:39

# Jelly, 6 bytes

-1 byte thanks to Erik the Outgolfer.

1x+2/Ƭ


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# Japt-R, 13 bytes

_ò mx}hUõÎü)â


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# Brachylog, 17 bytes

;1j₍ẹẉ₂{ġ₂+ᵐ}ⁱ.ẉȮ


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As horribly long as this is, I still feel a bit clever for using .ẉȮ: the obvious way to print something, then check if its length is 1 would be ẉ₂l1, ẉ₂~g, or ẉ₂≡Ȯ, where the ≡ in the last one is necessary because ẉ₂ unifies its input and output before it prints them, and Ȯ is pre-constrained to be a list of length 1, so the unification fails if the input is not a list of length 1. At the end of a predicate, this feature of ẉ₂ can be circumvented, however, by using the output variable instead of subscripting ẉ: .ẉȮ first unifies its input with the output variable, then prints the output variable, and only afterwards unifies the output variable with Ȯ.

# Stax, 10 bytes

Çë⌐ⁿ┤5π»Å╡


Run and debug it

## Procedure:

1. Generate 0-based range.
2. Repeatedly halve each element until all items are zero.
3. Calculate run-lengths for each unique array.

## Annotated Source:

r       main:[0 .. 5]
{{hmgu  main:[[0 .. 5], [0, 0, 1, 1, 2, 2], [0, 0, 0, 0, 1, 1], [0, 0, 0, 0, 0, 0]]
m:GJ    main:"1 1 1 1 1 1"


# JavaScript (V8), 109 bytes

f=n=>g(Array(n).fill(1));g=(a,i=1)=>{console.log(a);if(a[i]){for(;a[i];)a.splice(i-1,2,a[i-1]+a[i++]);g(a);}}


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# JavaScript, 55 bytes

f=(n,t=1,r=n)=>r>t?t+[,f(n,t,r-t)]:n>t?r+
+f(n,t+t):r


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This is basically the golfed version of following codes:

function f(n) {
var output = '';
t = 1;
for (t = 1; ; t *= 2) {
for (r = n; r > t; r -= t) {
output += t + ',';
}
output += r;
if (n <= t) break;
output += '\n';
}
return output;
}


# Charcoal, 19 bytes

ＮθＩＥ↨⊖⊗θ²Ｅ⪪Ｅθ¹Ｘ²κＬλ


Try it online! Link is to verbose version of code. Uses Charcoal's default output format, which is one number per line, with subarrays double-spaced from each other. Explanation:

Ｎθ                  Input n into a variable
θ            n
⊗             Doubled
⊖              Decremented
↨   ²           Converted to base 2 (i.e. ceil(log2(input)))
Ｅ                Map
Ｅθ¹      List of 1s of length n
⪪         Split into sublists of length
²    Literal 2
Ｘ     To power
κ   Loop index
Ｅ          Map over each sublist
Ｌλ Take the length
Ｉ                 Cast to string for implicit print


say$_="1 "x<>;say while s/(\d+) (\d+)/$1+$2/ge  Try it online! Output is space separated. # Perl 6, 38 bytes {1 xx$_,*.rotor(2,:partial)>>.sum...1}


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There's some shortcut to partial rotoring that I'm not remembering right now...

### Explanation:

{                                    }  # Anonymous code block
...    # Return a sequence
1 xx$_, # Starting with a list of 1s with input length * # Where each element is .rotor(2,:partial) # The previous list split into chunks of 2 or less >>.sum # And each chunk summed 1 # Until the list is length 1  # Haskell, 75 bytes g.pure g x|x!!0<2=[x]|1>0=(g$(\z->filter(0/=)[-div(-z)2,div z 2])=<<x)++[x]


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Works backwards from the list [n] until it reaches a list of just ones.

Going forwards, I could get 80 bytes using chunksof from Data.List.Split:

import Data.List.Split
f x=g$1<$[1..x]
g[n]=[[n]]
g x=x:(g$map sum$chunksOf 2 x)


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# Ohm v2, 8 bytes

@Dv·Ω2σΣ


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If output in scientific notation is allowed, otherwise:

# Ohm v2, 9 bytes

@Dv·Ω2σΣì


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• If the scientific notation numbers are actually a natural number type (such as floats) in Ohm then sure, it's reasonable. – Erik the Outgolfer Aug 22 at 12:48

# Gaia, 12 bytes

ċ)¦⟨:q2/Σ¦⟩ª


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ċ)¦		| generate array of n 1's (really, generate array of n 0's and increment each)
⟨      ⟩ª	| do the following until you get to a fixed point:
:q		| dup and print with a newline
2/	| split into groups of 2, with last group possibly being smaller
Σ¦	| take the sum