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
for n in range(1, int(input())+1):
    b=0
    a=[0]*(n+2)
    for i in range(1, n+1):
        if a[i]: continue
        for j in range(1, n+1):
            if a[j]: continue
            # we know z at this point
            d = i + j
            c = n - i**2 - j**2 - i*j
            try:
                k = int(math.sqrt(d**2 + 4*c) - d - 1)
                if k < 1: continue
                if n == (i**2 + j**2 + k**2 + i*j + j*k + i*k):
                    a[i] = True
                    a[j] = True
                    a[k] = True
                    if i == j and i == k and j == k:
                        b+= 1
                    elif i != j and i != k and j !=k:
                        b+=6
                    else:
                        b+=3
            except:
                pass
    print(b)