Problem:
Problem B: Lagrange's Four-Square Theorem

The fact that any positive integer has a representation as the sum of at most four positive
squares (i.e. squares of positive integers) is known as Lagrange’s Four-Square Theorem. The
first published proof of the theorem was given by Joseph-Louis Lagrange in 1770. Your mission
however is not to explain the original proof nor to discover a new proof but to show that the
theorem holds for some specific numbers by counting how many such possible representations
there are.

For a given positive integer n, you should report the number of all representations of n as the
sum of at most four positive squares. The order of addition does not matter, e.g. you should
consider 42 + 32 and 32 + 42 are the same representation.

For example, let’s check the case of 25. This integer has just three representations 12 +22 +22 +42 ,
32 + 42 , and 52 . Thus you should report 3 in this case. Be careful not to count 42 + 32 and
32 + 42 separately.

Input

The input is composed of at most 255 lines, each containing a single positive integer less than
215 , followed by a line containing a single zero. The last line is not a part of the input data.

Output

The output should be composed of lines, each containing a single integer. No other characters
should appear in the output.

The output integer corresponding to the input integer n is the number of all representations
of n as the sum of at most four positive squares.

Sample Input

1
25
2003
211
20007
0

Output for the Sample Input

1
3
48
7
738

Solution in C++14 (judge verdict: Accepted)

#include <bits/stdc++.h>
using namespace std;

int main() {
  cin.tie(nullptr);
  ios::sync_with_stdio(false);
  vector<int> res(1 << 15);
  vector<int> v;
  auto dfs = [&](auto&& self) -> void {
    int s = 0;
    for (int e : v) {
      s += e * e;
    }
    if (s < 1 << 15) {
      ++res[s];
    }
    if (v.size() == 4) {
      return;
    }
    for (int i = v.empty() ? 1 : v.back(); i < 200; ++i) {
      v.push_back(i);
      self(self);
      v.pop_back();
    }
  };
  dfs(dfs);
  int n;
  while (cin >> n, n) {
    cout << res[n] << '\n';
  }
}