Collection from a sequence that constitute a perfect square

Question

Given the sequence OEIS A033581, which is the infinite sequence, the n'th term (0-indexing) is given by the closed form formula 6 × n² .

Your task is to write code, which outputs all the subsets of the set of N first numbers in the sequence, such that the sum of the subset is a perfect square.

Rules

• The integer N is given as input.
• You cannot reuse a number already used in the sum. (that is, each number can appear in each subset at most once)
• Numbers used can be non-consecutive.
• Code with the least size wins.

Example

The given sequence is {0,6,24,54,96,...,15000}

One of the required subsets will be {6,24,294}, because

6+24+294 = 324 = 18^2

You need to find all such sets of all possible lengths in the given range.

Answer 1

05AB1E, 10 bytes

ݨn6*æʒOŲ

Try it online!

How?

ݨn6*æʒOŲ || Full program. I'll call the input N.

Ý          || 0-based inclusive range. Push [0, N] ∩ ℤ.
 ¨         || Remove the last element.
  n        || Square (element-wise).
   6*      || Multiply by 6.
     æ     || Powerset.
      ʒ    || Filter-keep those which satisfy the following:
       O   ||---| Their sum...
        Ų ||---| ... Is a perfect square?

Answer 2

Haskell, 114 104 103 86 bytes

f n=[x|x<-concat<$>mapM(\x->[[],[x*x*6]])[0..n-1],sum x==[y^2|y<-[0..],y^2>=sum x]!!0]

Thanks to Laikoni and Ørjan Johansen for most of the golfing! :)

Try it online!

The slightly more readable version:

--  OEIS A033581
ns=map((*6).(^2))[0..]

-- returns all subsets of a list (including the empty subset)
subsets :: [a] -> [[a]]
subsets[]=[[]]
subsets(x:y)=subsets y++map(x:)(subsets y)

-- returns True if the element is present in a sorted list
t#(x:xs)|t>x=t#xs|1<2=t==x

-- the function that returns the square subsets
f :: Int -> [[Int]]
f n = filter (\l->sum l#(map(^2)[0..])) $ subsets (take n ns)

Answer 3

Pyth, 12 bytes

-2 bytes thanks to Mr. Xcoder

fsI@sT2ym*6*

Try it online!

2 more bytes need to be added to remove [] and [0], but they seem like valid output to me!

---

Explanataion

    fsI@sT2ym*6*
    f                  filter
           y           the listified powerset of
            m*6*ddQ    the listified sequence {0,6,24,...,$input-th result}
        sT             where the sum of the sub-list
     sI@  2            is invariant over int parsing after square rooting

Answer 4

Clean, 145 ... 97 bytes

import StdEnv
@n=[[]:[[6*i^2:b]\\i<-[0..n-1],b<- @i]]
f=filter((\e=or[i^2==e\\i<-[0..e]])o sum)o@

Try it online!

Uses the helper function @ to generate the power set to n terms by concatenating each term of [[],[6*n^2],...] with each term of [[],[6*(n-1)*2],...] recursively, and in reverse order.

The partial function f is then composed (where -> denotes o composition) as:
apply @ -> take the elements where -> the sum -> is a square

Unfortunately it isn't possible to skip the f= and provide a partial function literal, because precedence rules require it have brackets when used inline.

Answer 5

JavaScript (ES7), 107 bytes

n=>[...Array(n)].reduce((a,_,x)=>[...a,...a.map(y=>[6*x*x,...y])],[[]]).filter(a=>eval(a.join`+`)**.5%1==0)

Demo

let f =

n=>[...Array(n)].reduce((a,_,x)=>[...a,...a.map(y=>[6*x*x,...y])],[[]]).filter(a=>eval(a.join`+`)**.5%1==0)

console.log(f(6).map(a => JSON.stringify(a)).join('\n'))
console.log(f(10).map(a => JSON.stringify(a)).join('\n'))

Commented

n =>                      // n = input
  [...Array(n)]           // generate a n-entry array
  .reduce((a, _, x) =>    // for each entry at index x:
    [                     //   update the main array a[] by:
      ...a,               //     concatenating the previous values with
      ...a.map(           //     new values built from the original ones
        y =>              //     where in each subarray y:
          [ 6 * x * x,    //       we insert a new element 6x² before
            ...y       ]  //       the original elements
      )                   //     end of map()
    ],                    //   end of array update
    [[]]                  //   start with an array containing an empty array
  )                       // end of reduce()
  .filter(a =>            // filter the results by keeping only elements for which:
    eval(a.join`+`) ** .5 //   the square root of the sum
    % 1 == 0              //   gives an integer
  )                       // end of filter()

Answer 6

Jelly, 12 bytes

Ḷ²6׌PSƲ$Ðf

Try it online!

Output is a list of subsets including 0s and the empty subset.

Answer 7

Wolfram Language (Mathematica), 49 bytes

Brute force approach

Select[Subsets[6Range[#]^2],Sqrt@Tr@#~Mod~1==0&]&

Try it online!

Answer 8

Japt, 15 bytes

ò_²*6Ãà k_x ¬u1

Try it

---

Explanation

Generate on array of integers from 0 to input (ò) and pass each through a function (_ Ã), squaring it (²) & mutiplying by 6 (*6). Get all the combinations of that array (à) and remove those that return truthy (k) when passed through a function (_) that adds their elements (x), gets the square root of the result (¬) and mods that by 1 (u1)