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 (1.42.0) (judge verdict: Accepted)

use proconio::{input, fastout, marker::Usize1};
use std::collections::{VecDeque, HashSet};

#[fastout]
fn main() {
    input! {
        n: usize, x: Usize1, y: Usize1,
    }

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

    let mut set: HashSet<usize> = HashSet::new();
    let mut ans = vec![0; n];
    for i in 0..n-1 {
        let mut queue: VecDeque<(usize, usize)> = VecDeque::new();
        queue.push_back((i, 0));
        while let Some((j, k)) = queue.pop_front() {
            let a = i + n * j;
            if set.contains(&a) { continue; }
            set.insert(a);
            if j > i { ans[k] += 1; }
            for l in &graph[j] { queue.push_back((*l, k+1)); }
        }
    }

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