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: Accepted)

// The main code is at the very bottom.

#[allow(unused_imports)]
use {
  lib::byte::ByteChar,
  std::cell::{Cell, RefCell},
  std::cmp::{
    self,
    Ordering::{self, *},
    Reverse,
  },
  std::collections::*,
  std::convert::identity,
  std::fmt::{self, Debug, Display, Formatter},
  std::io::prelude::*,
  std::iter::{self, FromIterator},
  std::marker::PhantomData,
  std::mem,
  std::num::Wrapping,
  std::ops::{Range, RangeFrom, RangeInclusive, RangeTo, RangeToInclusive},
  std::process,
  std::rc::Rc,
  std::thread,
  std::time::{Duration, Instant},
  std::{char, f32, f64, i128, i16, i32, i64, i8, isize, str, u128, u16, u32, u64, u8, usize},
};

#[allow(unused_imports)]
#[macro_use]
pub mod lib {
  pub mod byte {
    pub use self::byte_char::*;

    mod byte_char {
      use std::error::Error;
      use std::fmt::{self, Debug, Display, Formatter};
      use std::str::FromStr;

      #[derive(Clone, Copy, Default, PartialEq, Eq, PartialOrd, Ord, Hash)]
      #[repr(transparent)]
      pub struct ByteChar(pub u8);

      impl Debug for ByteChar {
        fn fmt(&self, f: &mut Formatter) -> fmt::Result {
          write!(f, "b'{}'", self.0 as char)
        }
      }

      impl Display for ByteChar {
        fn fmt(&self, f: &mut Formatter) -> fmt::Result {
          write!(f, "{}", self.0 as char)
        }
      }

      impl FromStr for ByteChar {
        type Err = ParseByteCharError;

        fn from_str(s: &str) -> Result<ByteChar, ParseByteCharError> {
          match s.as_bytes().len() {
            1 => Ok(ByteChar(s.as_bytes()[0])),
            0 => Err(ParseByteCharErrorKind::EmptyStr.into()),
            _ => Err(ParseByteCharErrorKind::TooManyBytes.into()),
          }
        }
      }

      #[derive(Clone, Copy, PartialEq, Eq, Hash, Debug)]
      pub struct ParseByteCharError {
        kind: ParseByteCharErrorKind,
      }

      impl Display for ParseByteCharError {
        fn fmt(&self, f: &mut Formatter) -> fmt::Result {
          f.write_str(match self.kind {
            ParseByteCharErrorKind::EmptyStr => "empty string",
            ParseByteCharErrorKind::TooManyBytes => "too many bytes",
          })
        }
      }

      impl Error for ParseByteCharError {}

      #[derive(Clone, Copy, PartialEq, Eq, Hash, Debug)]
      enum ParseByteCharErrorKind {
        EmptyStr,
        TooManyBytes,
      }

      impl From<ParseByteCharErrorKind> for ParseByteCharError {
        fn from(kind: ParseByteCharErrorKind) -> ParseByteCharError {
          ParseByteCharError { kind }
        }
      }
    }
  }

  pub mod io {
    pub use self::scanner::*;

    mod scanner {
      use std::io::{self, BufRead};
      use std::iter;
      use std::str::FromStr;

      #[derive(Debug)]
      pub struct Scanner<R> {
        reader: R,
        buf: String,
        pos: usize,
      }

      impl<R: BufRead> Scanner<R> {
        pub fn new(reader: R) -> Self {
          Scanner {
            reader,
            buf: String::new(),
            pos: 0,
          }
        }

        pub fn next(&mut self) -> io::Result<&str> {
          let start = loop {
            match self.rest().find(|c| c != ' ') {
              Some(i) => break i,
              None => self.fill_buf()?,
            }
          };
          self.pos += start;
          let len = self.rest().find(' ').unwrap_or(self.rest().len());
          let s = &self.buf[self.pos..][..len]; // self.rest()[..len]
          self.pos += len;
          Ok(s)
        }

        pub fn parse_next<T>(&mut self) -> io::Result<Result<T, T::Err>>
        where
          T: FromStr,
        {
          Ok(self.next()?.parse())
        }

        pub fn parse_next_n<T>(&mut self, n: usize) -> io::Result<Result<Vec<T>, T::Err>>
        where
          T: FromStr,
        {
          iter::repeat_with(|| self.parse_next()).take(n).collect()
        }

        pub fn map_next_bytes<T, F>(&mut self, mut f: F) -> io::Result<Vec<T>>
        where
          F: FnMut(u8) -> T,
        {
          Ok(self.next()?.bytes().map(&mut f).collect())
        }

        pub fn map_next_bytes_n<T, F>(&mut self, n: usize, mut f: F) -> io::Result<Vec<Vec<T>>>
        where
          F: FnMut(u8) -> T,
        {
          iter::repeat_with(|| self.map_next_bytes(&mut f))
            .take(n)
            .collect()
        }

        fn rest(&self) -> &str {
          &self.buf[self.pos..]
        }

        fn fill_buf(&mut self) -> io::Result<()> {
          self.buf.clear();
          self.pos = 0;
          let read = self.reader.read_line(&mut self.buf)?;
          if read == 0 {
            return Err(io::ErrorKind::UnexpectedEof.into());
          }
          if *self.buf.as_bytes().last().unwrap() == b'\n' {
            self.buf.pop();
          }
          Ok(())
        }
      }
    }
  }
}

#[allow(unused_macros)]
macro_rules! eprint {
  ($($arg:tt)*) => {
    if cfg!(debug_assertions) {
      std::eprint!($($arg)*)
    }
  };
}
#[allow(unused_macros)]
macro_rules! eprintln {
  ($($arg:tt)*) => {
    if cfg!(debug_assertions) {
      std::eprintln!($($arg)*)
    }
  };
}
#[allow(unused_macros)]
macro_rules! dbg {
  ($($arg:tt)*) => {
    if cfg!(debug_assertions) {
      std::dbg!($($arg)*)
    } else {
      ($($arg)*)
    }
  };
}

const CUSTOM_STACK_SIZE_MIB: Option<usize> = Some(128);
const INTERACTIVE: bool = false;

fn main() -> std::io::Result<()> {
  match CUSTOM_STACK_SIZE_MIB {
    Some(stack_size_mib) => std::thread::Builder::new()
      .name("run_solver".to_owned())
      .stack_size(stack_size_mib * 1024 * 1024)
      .spawn(run_solver)?
      .join()
      .unwrap(),
    None => run_solver(),
  }
}

fn run_solver() -> std::io::Result<()> {
  let stdin = std::io::stdin();
  let reader = stdin.lock();
  let stdout = std::io::stdout();
  let writer = stdout.lock();
  macro_rules! with_wrapper {
    ($($wrapper:expr)?) => {{
      let mut writer = $($wrapper)?(writer);
      solve(reader, &mut writer)?;
      writer.flush()
    }};
  }
  if cfg!(debug_assertions) || INTERACTIVE {
    with_wrapper!()
  } else {
    with_wrapper!(std::io::BufWriter::new)
  }
}

fn solve<R, W>(reader: R, mut writer: W) -> std::io::Result<()>
where
  R: BufRead,
  W: Write,
{
  let mut _scanner = lib::io::Scanner::new(reader);
  #[allow(unused_macros)]
  macro_rules! scan {
    ($T:ty) => {
      _scanner.parse_next::<$T>()?.unwrap()
    };
    ($($T:ty),+) => {
      ($(scan!($T)),+)
    };
    ($T:ty; $n:expr) => {
      _scanner.parse_next_n::<$T>($n)?.unwrap()
    };
    ($($T:ty),+; $n:expr) => {
      iter::repeat_with(|| -> std::io::Result<_> { Ok(($(scan!($T)),+)) })
        .take($n)
        .collect::<std::io::Result<Vec<_>>>()?
    };
  }
  #[allow(unused_macros)]
  macro_rules! scan_bytes_map {
    ($f:expr) => {
      _scanner.map_next_bytes($f)?
    };
    ($f:expr; $n:expr) => {
      _scanner.map_next_bytes_n($n, $f)?
    };
  }
  #[allow(unused_macros)]
  macro_rules! print {
    ($($arg:tt)*) => {
      write!(writer, $($arg)*)?
    };
  }
  #[allow(unused_macros)]
  macro_rules! println {
    ($($arg:tt)*) => {
      writeln!(writer, $($arg)*)?
    };
  }
  #[allow(unused_macros)]
  macro_rules! answer {
    ($($arg:tt)*) => {{
      println!($($arg)*);
      return Ok(());
    }};
  }
  {
    let n = scan!(u64);

    let mut cnt = vec![0; (n + 1) as usize];
    for x in 1u64..1000 {
      for y in 1u64..1000 {
        for z in 1u64..1000 {
          if let Some(cnt) = cnt.get_mut((x * x + y * y + z * z + x * y + y * z + z * x) as usize) {
            *cnt += 1;
          }
        }
      }
    }

    for i in 1..=n {
      println!("{}", cnt[i as usize]);
    }
  }
  #[allow(unreachable_code)]
  Ok(())
}