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 Python (3.8.2) (judge verdict: Time Limit Exceeded)

def f(n):
    pairs = set()
    for x in range(1, 45):
        for y in range(x, 45):
            temp = - 3 * (x + y) ** 2 + 4 * n + 4 * x * y
            if temp < 0:
                continue
            temp2 = int(temp ** .5)
            if temp2 * temp2 != temp:
                continue
            temp3 = - (x + y) + temp2
            temp4 = - (x + y) - temp2
            z1 = temp3 // 2
            z2 = temp4 // 2
            if z1 >= 1 and temp3 % 2 == 0:
                key = tuple(sorted((x, y, z1)))
                if key in pairs:
                    continue
                pairs.add(key)
            if z2 >= 1 and temp4 % 2 == 0:
                key = tuple(sorted((x, y, z2)))
                if key in pairs:
                    continue
                pairs.add(key)
    res = 0
    for x, y, z in pairs:
        if x == y == z:
            res += 1
        elif x == y or y == z or z == x:
            res += 3
        else:
            res += 6
    return res


N = int(input())
for i in range(1, N + 1):
    print(f(i))