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 PyPy3 (7.3.0) (judge verdict: Accepted)

def f(x, y, z):
    return (x+y+z)**2 - x*y - y*z - z*x


N = int(input())
for i in range(1, N+1):
    ans = 0
    for x in range(1, int(i**0.5)+2):
        for y in range(x, int(i**0.5)+2):
            if f(x, y, y-1) >= i:
                break
            l = y - 1
            r = int(i**0.5) + 1
            while r >= l:
                z = (r+l) // 2
                if f(x, y, z) == i:
                    if x < y:
                        if x == z or y == z:
                            ans += 3
                        else:
                            ans += 6
                    else:
                        if z == y:
                            ans += 1
                        else:
                            ans += 3
                if f(x, y, z) > i:
                    r = z - 1
                else:
                    l = z + 1

    print(ans)