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 Rust (1.42.0) (judge verdict: Time Limit Exceeded)

#![allow(non_snake_case)]
#![allow(unused_imports)]

use proconio::{input, fastout};
use proconio::marker::*;
use whiteread::parse_line;
use std::collections::*;
use num::*;
use num_traits::*;
use superslice::*;
use std::ops::*;

use itertools::Itertools;
use itertools_num::ItertoolsNum;

#[fastout]
fn main() {
    input!{
        n: usize,
    }
    let mut ans_vec = vec![];
    for i in 1..=n {
        let mut ans = 0;
        let x_bound = (i as f64).sqrt() as usize;
        // dbg!(&x_bound);
        for x in 1..=x_bound {
            if i <= x * x + x {
                continue;
            }

            let y_bound = ((i - x*x) as f64).sqrt() as usize;

            for y in 1..=y_bound {
                if i <= x * x + y * y + x * y + x + y {
                    continue;
                }

                let b24c =  (x + y).pow(2) + 4 * (i - x.pow(2) - y.pow(2) - x * y);
                if !num::rational::Ratio::from_float((b24c as f64).sqrt()).unwrap().is_integer() {
                    continue;
                }

                if (x + y) as f64 > (b24c as f64).sqrt() {
                    continue;
                }

                ans += 1;
            }
        }
        ans_vec.push(ans);
    }
    println!("{}", ans_vec.iter().map(|x| x.to_string()).join("\n"))
}