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(unused_imports)]
use itertools::Itertools;
#[allow(unused_imports)]
use itertools_num::ItertoolsNum;
#[allow(unused_imports)]
use std::cmp;
#[allow(unused_imports)]
use std::iter;
#[allow(unused_imports)]
use superslice::*;

fn run() {
    let (r, w) = (std::io::stdin(), std::io::stdout());
    let mut sc = IO::new(r.lock(), w.lock());

    let n: usize = sc.read();

    for i in 1..=n {
        let mut ans = 0;
        for x in 1..=i {
            if x * x > i {
                break;
            }
            for y in 1..=i {
                if y * y > n {
                    break;
                }
                for z in 1..=i {
                    if z * z > n {
                        break;
                    }
                    if x * x + y * y + z * z + x * y + y * z + z * x == i {
                        ans += 1;
                    }
                }
            }
        }
        println!("{}", ans);
    }
}

fn main() {
    std::thread::Builder::new()
        .name("run".into())
        .stack_size(256 * 1024 * 1024)
        .spawn(run)
        .unwrap()
        .join()
        .unwrap();
}

pub struct IO<R, W: std::io::Write>(R, std::io::BufWriter<W>);

impl<R: std::io::Read, W: std::io::Write> IO<R, W> {
    pub fn new(r: R, w: W) -> IO<R, W> {
        IO(r, std::io::BufWriter::new(w))
    }
    pub fn write<S: std::ops::Deref<Target = str>>(&mut self, s: S) {
        use std::io::Write;
        self.1.write(s.as_bytes()).unwrap();
    }
    pub fn read<T: std::str::FromStr>(&mut self) -> T {
        use std::io::Read;
        let buf = self
            .0
            .by_ref()
            .bytes()
            .map(|b| b.unwrap())
            .skip_while(|&b| b == b' ' || b == b'\n' || b == b'\r' || b == b'\t')
            .take_while(|&b| b != b' ' && b != b'\n' && b != b'\r' && b != b'\t')
            .collect::<Vec<_>>();
        unsafe { std::str::from_utf8_unchecked(&buf) }
            .parse()
            .ok()
            .expect("Parse error.")
    }
    pub fn read_vec<T: std::str::FromStr>(&mut self, n: usize) -> Vec<T> {
        (0..n).map(|_| self.read()).collect()
    }
    pub fn read_pairs<T: std::str::FromStr>(&mut self, n: usize) -> Vec<(T, T)> {
        (0..n).map(|_| (self.read(), self.read())).collect()
    }
    pub fn read_pairs_1_indexed(&mut self, n: usize) -> Vec<(usize, usize)> {
        (0..n)
            .map(|_| (self.read::<usize>() - 1, self.read::<usize>() - 1))
            .collect()
    }
    pub fn read_chars(&mut self) -> Vec<char> {
        self.read::<String>().chars().collect()
    }
    pub fn read_char_grid(&mut self, n: usize) -> Vec<Vec<char>> {
        (0..n).map(|_| self.read_chars()).collect()
    }
    pub fn read_matrix<T: std::str::FromStr>(&mut self, n: usize, m: usize) -> Vec<Vec<T>> {
        (0..n)
            .map(|_| (0..m).map(|_| self.read()).collect())
            .collect()
    }
}