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):
    if n < 2:
        return 0
    lim1 = int((n - 2) ** 0.5 - 1) + 1
    lim2 = int((-2 + (4 - 12 * (1 - n)) ** 0.5) / 6) + 1
    lim3 = int(n / 6) + 1

    result = set()
    for x in range(1, lim3):
        for y in range(1, lim2):
            if x > y:
                continue
            for z in range(1, lim1):
                if y > z:
                    continue
                if x ** 2 + y ** 2 + z ** 2 + x * y + y * z + z * x == n:
                    tmp = set([x, y, z])
                    if tmp not in result:
                        result.add(tuple(tmp))
    count = 0
    for s in result:
        if len(s) == 1:
            count += 1
        elif len(s) == 2:
            count += 3
        else:
            count += 6
    return count


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