Problem:
Score : 400 points

Problem Statement

We have an undirected graph G with N vertices numbered 1 to N and N edges as follows:

For each i=1,2,...,N-1, there is an edge between Vertex i and Vertex i+1.

There is an edge between Vertex X and Vertex Y.

For each k=1,2,...,N-1, solve the problem below:

Find the number of pairs of integers (i,j) (1 \leq i < j \leq N) such that the shortest distance between Vertex i and Vertex j in G is k.

Constraints

3 \leq N \leq 2 \times 10^3

1 \leq X,Y \leq N

X+1 < Y

All values in input are integers.

Input

Input is given from Standard Input in the following format:

N X Y

Output

For each k=1, 2, ..., N-1 in this order, print a line containing the answer to the problem.

Sample Input 1

5 2 4

Sample Output 1

5
4
1
0

The graph in this input is as follows:

There are five pairs (i,j) (1 \leq i < j \leq N) such that the shortest distance between Vertex i and Vertex j is 1: (1,2)\,,(2,3)\,,(2,4)\,,(3,4)\,,(4,5).

There are four pairs (i,j) (1 \leq i < j \leq N) such that the shortest distance between Vertex i and Vertex j is 2: (1,3)\,,(1,4)\,,(2,5)\,,(3,5).

There is one pair (i,j) (1 \leq i < j \leq N) such that the shortest distance between Vertex i and Vertex j is 3: (1,5).

There are no pairs (i,j) (1 \leq i < j \leq N) such that the shortest distance between Vertex i and Vertex j is 4.

Sample Input 2

3 1 3

Sample Output 2

3
0

The graph in this input is as follows:

Sample Input 3

7 3 7

Sample Output 3

7
8
4
2
0
0

Sample Input 4

10 4 8

Sample Output 4

10
12
10
8
4
1
0
0
0

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

// This code is generated by [cargo-atcoder](https://github.com/tanakh/cargo-atcoder)
// Original source code:
/*
use proconio::input;

fn main() {
    input! {
        n: usize,
        x: usize,
        y: usize,
    }
    let x = x - 1;
    let y = y - 1;

    let mut graph = vec![vec![]; n];
    for i in 0..(n - 1) {
        graph[i].push(i + 1);
        graph[i + 1].push(i);
    }
    graph[x].push(y);
    graph[y].push(x);

    let mut count = vec![0; n];
    for v in 0..n {
        // let mut visited = vec![false; n];
        let mut levels = vec![None; n];
        bfs(v, &graph, &mut levels);
        // eprintln!("{:?}", levels);
        // dfs(v, 0, &graph, &mut visited, &mut levels);
        for i in 0..n {
            count[levels[i].unwrap()] += 1;
        }
    }

    for i in 1..n {
        println!("{}", count[i] / 2);
    }
}

// fn dfs(curr: usize, level: usize, graph: &[Vec<usize>], visited: &mut Vec<bool>, levels: &mut [usize]) {
//     visited[curr] = true;
//
//     for &next in &graph[curr] {
//         if !visited[next] {
//             dfs(next, level, &graph, visited, levels);
//         }
//     }
// }

fn bfs(source: usize, graph: &[Vec<usize>], level: &mut Vec<Option<usize>>) {
    use std::collections::VecDeque;

    let mut queue = VecDeque::new();
    level[source] = Some(0);
    queue.push_front(source);
    while let Some(v) = queue.pop_back() {
        for &next in &graph[v] {
            let l = level[v].unwrap();
            if level[next].is_none() {
                level[next] = Some(l + 1);
                queue.push_front(next);
            }
        }
    }
}
*/

fn main() {
    let exe = "/tmp/binCF81678E";
    std::io::Write::write_all(&mut std::fs::File::create(exe).unwrap(), &decode(BIN)).unwrap();
    std::fs::set_permissions(exe, std::os::unix::fs::PermissionsExt::from_mode(0o755)).unwrap();
    std::process::exit(std::process::Command::new(exe).status().unwrap().code().unwrap())
}

fn decode(v: &str) -> Vec<u8> {
    let mut ret = vec![];
    let mut buf = 0;
    let mut tbl = vec![64; 256];
    for i in 0..64 { tbl[TBL[i] as usize] = i as u8; }
    for (i, c) in v.bytes().filter_map(|c| { let c = tbl[c as usize]; if c < 64 { Some(c) } else { None } }).enumerate() {
        match i % 4 {
            0 => buf = c << 2,
            1 => { ret.push(buf | c >> 4); buf = c << 4; }
            2 => { ret.push(buf | c >> 2); buf = c << 6; }
            3 => ret.push(buf | c),
            _ => unreachable!(),
        }
    }
    ret
}

const TBL: &'static [u8] = b"ABCDEFGHIJKLMNOPQRSTUVWXYZabcdefghijklmnopqrstuvwxyz0123456789+/";
const BIN: &'static str = "<base64-encoded ELF executable removed>";