My code

… was written in C++11 and can be compiled with G++, Clang++, Visual C++. You can download it, too. Or just jump to my GitHub repository.

       
#include <iostream>
       
#include <vector>
       
 
       
int main()
       
{
       
  unsigned int maxSet = 100;
       
  unsigned int choose =  50;
       
  std::cin >> maxSet >> choose;
       
 
       
  // all squares
       
  std::vector<unsigned int> values;
       
  for (unsigned int i = 1; i <= maxSet; i++)
       
    values.push_back(i*i);
       
 
       
  std::vector<std::vector<bool>> reachable (choose + 1);
       
  std::vector<std::vector<bool>> duplicates(choose + 1);
       
  for (size_t i = 0; i < reachable.size(); i++)
       
  {
       
    // heuristic to allocate only as much memory as needed
       
    size_t maxSize = maxSet*maxSet * i + 1;
       
    if (maxSize > 300000) // somewhat below 300000 different sums
       
      maxSize = 300000;
       
    reachable [i].resize(maxSize, false);
       
    duplicates[i].resize(maxSize, false);
       
  }
       
 
       
  // the empty set is always possible
       
  reachable[0][0] = true;
       
 
       
  // for each square ...
       
  for (auto add : values)
       
  {
       
    // try to append it to any valid set
       
    for (unsigned int numElements = choose; numElements > 0; numElements--)
       
    {
       
      // try every valid sum
       
      for (size_t sum = 0; sum < reachable[numElements - 1].size(); sum++)
       
      {
       
        // sum must be possible
       
        if (!reachable[numElements - 1][sum])
       
          continue;
       
 
       
        // append number to previous sum
       
        auto current = add + sum;
       
 
       
        // "sum" was a duplicate or "current" already exists ?
       
        if (duplicates[numElements - 1][sum] ||
       
            reachable[numElements][current])
       
        {
       
          // new sum is reachable but invalid
       
          reachable [numElements][current] = true;
       
          duplicates[numElements][current] = true;
       
        }
       
        else
       
        {
       
          // yes, new unique sum
       
          reachable[numElements][current] = true;
       
        }
       
      }
       
    }
       
  }
       
 
       
  // count all unique sums
       
  unsigned int result = 0;
       
  for (size_t sum = 0; sum < reachable[choose].size(); sum++)
       
    if (reachable[choose][sum] && !duplicates[choose][sum])
       
      result += sum;
       
 
       
  std::cout << result << std::endl;
       
  return 0;
       
}

This solution contains 9 empty lines, 12 comments and 2 preprocessor commands.