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 PHP (7.4.4) (judge verdict: Accepted)

<?php
list($n) = ints();
$max = (int) (sqrt($n / 3) + 1);
for ($i = 1; $i <= $n; ++$i) {
    $ans[$i] = 0;
    for ($x = 1; $x <= $max; ++$x) {
        for ($y = $x; $y <= $max; ++$y) {
            if ($x ** 2 + $y ** 2 + 1 + $x * $y + $x + $y > $n) break 2;
            $t = ($x + $y) ** 2 - 4 * ($x ** 2 + $y ** 2 + $x * $y - $i);
            if ($t > 0 && ((int) (sqrt($t))) ** 2 === $t && (int) sqrt($t) >= 2 + $x + $y) {
                $z = intdiv(((int) sqrt($t) - ($x + $y)), 2);
                if ($z < $y) break;
                $cnt = count(array_unique([$x, $y, $z]));
                if ($cnt === 1) {
                    ++$ans[$i];
                } elseif ($cnt === 2) {
                    $ans[$i] += 3;
                } else {
                    $ans[$i] += 6;
                }
            }
        }
    }
}
echo implode(PHP_EOL, $ans);
function ints()
{
    return array_map('intval', explode(' ', trim(fgets(STDIN))));
}