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)

import math

n = int(input())
maxn = math.ceil(n**0.5)

for i in range(1, n+1, 1):
    maxn = math.ceil((i/2.0)**0.5)
    cnt = 0
    for x in range(1, maxn+1):
        if x**2 > i:
            break
        for y in range(x, maxn+1):
            if x**2 + y**2 + x* y > i:
                break
            for z in range(y, maxn+1):
                if x**2 + y**2 + z**2 + x*y + y*z + z*x == i:
                    if x == y and y == z:
                        cnt += 1
                    elif x != y and y != z:
                        cnt += 6
                    else:
                        cnt += 3
                elif x**2 + y**2 + z**2 + x*y + y*z + z*x > i:
                    break
    print(cnt)