Problem:
Score : 400 points

Problem Statement

There is a knight - the chess piece - at the origin (0, 0) of a two-dimensional grid.

When the knight is at the square (i, j), it can be moved to either (i+1,j+2) or (i+2, j+1).

In how many ways can the knight reach the square (X, Y)?

Find the number of ways modulo 10^9 + 7.

Constraints

1 \leq X \leq 10^6

1 \leq Y \leq 10^6

All values in input are integers.

Input

Input is given from Standard Input in the following format:

X Y

Output

Print the number of ways for the knight to reach (X, Y) from (0, 0), modulo 10^9 + 7.

Sample Input 1

3 3

Sample Output 1

2

There are two ways: (0,0) \to (1,2) \to (3,3) and (0,0) \to (2,1) \to (3,3).

Sample Input 2

2 2

Sample Output 2

0

The knight cannot reach (2,2).

Sample Input 3

999999 999999

Sample Output 3

151840682

Print the number of ways modulo 10^9 + 7.

Solution in Rust (1.15.1) (judge verdict: Accepted)

use std::cmp::min;

const MOD: i64 = 1000000007;


fn main() {
    let (x, y) = read_tuple::<usize>();
    let c = Comb::new(1e6 as i64 as usize + 1, MOD);

    let mut ans = 0;
    for s in 0..min(x+1, y/2+1) {
        let t = y - 2 * s;
        if (x - s) % 2 == 0 && t == (x - s) / 2 {
            ans += c.nck(s + t, s);
            ans %= MOD;
        }
    }

    println!("{}", ans);
}

/// for nCk
pub struct Comb {
    max: usize,
    m: i64,
    fac: Vec<i64>,
    finv: Vec<i64>,
}

impl Comb {
    /// initialize table 
    /// max: maximum that n as nCk takes
    /// m: MOD, a prime number
    /// O(n)
    pub fn new(max: usize, m: i64) -> Comb {
        let n = max;
        let mut fac = Vec::with_capacity(n);
        fac.push(1);
        fac.push(1);
        let mut finv = Vec::with_capacity(n);
        finv.push(1);
        finv.push(1);
        let mut inv = Vec::with_capacity(n);
        inv.push(1);
        inv.push(1);

        for i in 2..n {
            let i = i as i64;
            let tmp = fac.last().unwrap() * i % m;
            fac.push(tmp);
            let tmp = m - inv[(m % i) as usize] * (m / i) % m;
            inv.push(tmp);
            let tmp = finv.last().unwrap() * inv.last().unwrap() % m;
            finv.push(tmp);
        }

        Comb {
            max: max,
            m: m,
            fac: fac,
            finv: finv
        }
    }

    /// nCk
    pub fn nck(&self, n: usize, k: usize) -> i64 {
        // since n, k are usize, do not need to check `n < 0 || k < 0`
        if n < k || n > self.max {
            panic!("Invalid query (n, k) = ({}, {})", n, k);
        } else {
            self.fac[n] * (self.finv[k] * self.finv[n - k] % self.m) % self.m
        }
    }
}


fn read_vec<T: std::str::FromStr>() -> Vec<T> {
    let mut s = String::new();
    std::io::stdin().read_line(&mut s).ok();
    s.trim().split_whitespace()
        .map(|e| e.parse().ok().unwrap()).collect()
}
fn read_tuple<T: std::str::FromStr + Copy>() -> (T, T) {
    let temp = read_vec();
    (temp[0], temp[1])
}