Problem:
Score : 300 points

Problem Statement

Let f(n) be the number of triples of integers (x,y,z) that satisfy both of the following conditions:

1 \leq x,y,z

x^2 + y^2 + z^2 + xy + yz + zx = n

Given an integer N, find each of f(1),f(2),f(3),\ldots,f(N).

Constraints

All values in input are integers.

1 \leq N \leq 10^4

Input

Input is given from Standard Input in the following format:

N

Output

Print N lines. The i-th line should contain the value f(i).

Sample Input 1

20

Sample Output 1

0
0
0
0
0
1
0
0
0
0
3
0
0
0
0
0
3
3
0
0

For n=6, only (1,1,1) satisfies both of the conditions. Thus, f(6) = 1.

For n=11, three triples, (1,1,2), (1,2,1), and (2,1,1), satisfy both of the conditions. Thus, f(6) = 3.

For n=17, three triples, (1,2,2), (2,1,2), and (2,2,1), satisfy both of the conditions. Thus, f(17) = 3.

For n=18, three triples, (1,1,3), (1,3,1), and (3,1,1), satisfy both of the conditions. Thus, f(18) = 3.

Solution in Ruby (2.7.1) (judge verdict: Time Limit Exceeded)

n = gets.chomp.to_i

def xyz(ary, n)
  xy = ary[0] * ary[1]
  yz = ary[1] * ary[2]
  zx = ary[2] * ary[0]
  res = ary[0]**2 + ary[1]**2 + ary[2]**2 + xy + yz + zx

  res == n ? true : false
end

def xyz_function(k)
  x, y, z = 0
  cnt = 0
  return cnt if k < 3
  ary = [*1..k]
  ary_comb = ary.repeated_permutation(3).to_a
  ary_comb_d1 = ary_comb.delete_if{|ay| ay.sum > k}
  ary_comb_d2 = ary_comb_d1.delete_if{|ay| 
    ay[0]**2 + ay[1]**2 + ay[2]**2 > k
  }
  ary_comb_d2.each{|ary_f|
    if xyz(ary_f, k)
      cnt += 1
    end
  }

  cnt > 0 ? cnt : 0
end

1.upto(n) {|k|
  puts xyz_function(k)
}