Buggy python submission (verdict: Time Limit Exceeded):
# -*- coding: utf-8 -*-
import math

# def f2(x, y, z, n):
#     return x*x+y*y+z*z+x*y+y*z+z*x

def f(x, y, z, n):
    if x*x+y*y+z*z+x*y+y*z+z*x == n:
        return True
    else:
        return False
def main():
    N = int(input())
    for n in range(1, N + 1):
        cnt = 0
        for x in range(1, 100):
            if x * x > n:
                break
            for y in range(x, 100):
                if x*x + y*y > n:
                    break
                for z in range(y, 100):
                    if x*x + y*y + z*z > n:
                        break
                    if f(x, y, z, n):
                        # print(x, end=' ')
                        # print(y, end=' ')
                        # print(z)
                        if x == y == z:
                            cnt += 1
                        elif x != y and y != z and z != x:
                            cnt += 6
                        else:
                            cnt += 3
        # print(n, end=': ')
        print(cnt)
    return
if __name__ == '__main__':
    main()

Fixed submission:
# -*- coding: utf-8 -*-
import math

# def f2(x, y, z, n):
#     return x*x+y*y+z*z+x*y+y*z+z*x

def f(x, y, z, n):
    if x*x+y*y+z*z+x*y+y*z+z*x == n:
        return True
    else:
        return False
def main():
    N = int(input())
    for n in range(1, N + 1):
        cnt = 0
        for x in range(1, 100):
            if x * x > n:
                break
            for y in range(x, 100):
                if x*x + y*y > n:
                    break
                for z in range(y, 100):
                    if x*x + y*y + z*z > n:
                        break
                    if f(x, y, z, n):
                        # print(x, end=' ')
                        # print(y, end=' ')
                        # print(z)
                        if x == y == z:
                            cnt += 1
                        elif x != y and y != z and z != x:
                            cnt += 6
                        else:
                            cnt += 3
        # print(n, end=': ')
        print(cnt)
    return
if __name__ == '__main__':
    main()

Buggy python submission (verdict: Time Limit Exceeded):
# -*- coding: utf-8 -*-
import math

# def f2(x, y, z, n):
#     return x*x+y*y+z*z+x*y+y*z+z*x

def f(x, y, z, n):
    if x*x+y*y+z*z+x*y+y*z+z*x == n:
        return True
    else:
        return False
def main():
    N = int(input())
    for n in range(1, N + 1):
        cnt = 0
        for x in range(1, 100):
            if x * x > n:
                break
            for y in range(x, 100):
                if x*x + y*y > n:
                    break
                for z in range(y, 100):
                    if x*x + y*y + z*z > n:
                        break
                    if f(x, y, z, n):
                        # print(x, end=' ')
                        # print(y, end=' ')
                        # print(z)
                        if x == y == z:
                            cnt += 1
                        else:
                            cnt += 3
        # print(n, end=': ')
        print(cnt)
    return
if __name__ == '__main__':
    main()

Fixed submission:
# -*- coding: utf-8 -*-
import math

# def f2(x, y, z, n):
#     return x*x+y*y+z*z+x*y+y*z+z*x

def f(x, y, z, n):
    if x*x+y*y+z*z+x*y+y*z+z*x == n:
        return True
    else:
        return False
def main():
    N = int(input())
    for n in range(1, N + 1):
        cnt = 0
        for x in range(1, 100):
            if x * x > n:
                break
            for y in range(x, 100):
                if x*x + y*y > n:
                    break
                for z in range(y, 100):
                    if x*x + y*y + z*z > n:
                        break
                    if f(x, y, z, n):
                        # print(x, end=' ')
                        # print(y, end=' ')
                        # print(z)
                        if x == y == z:
                            cnt += 1
                        elif x != y and y != z and z != x:
                            cnt += 6
                        else:
                            cnt += 3
        # print(n, end=': ')
        print(cnt)
    return
if __name__ == '__main__':
    main()