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: Time Limit Exceeded)

fn main() {
    proconio::input! {
        n: usize,
        x: usize,
        y: usize,
    }
    let mut hm = std::collections::HashMap::new();
    for i in 1..n {
        hm.entry(i - 1).or_insert(vec![]).push(i + 1 - 1);
        hm.entry(i + 1 - 1).or_insert(vec![]).push(i - 1);
    }
    hm.entry(x - 1).or_insert(vec![]).push(y - 1);
    hm.entry(y - 1).or_insert(vec![]).push(x - 1);
    for k in 1..n {
        let mut count = 0usize;
        for i in 0..n {
            for j in i + 1..n {
                let mut v = vec![std::usize::MAX; n];
                v[i] = 0;
                let mut deque = std::collections::VecDeque::new();
                deque.push_back((i, 0));
                'outer: while let Some((index, distance)) = deque.pop_front() {
                    if distance > k {
                        break;
                    }
                    for &neighbor in hm.get(&index).unwrap() {
                        if v[neighbor] == std::usize::MAX {
                            v[neighbor] = distance + 1;
                            if neighbor == j {
                                break 'outer;
                            }
                            deque.push_back((neighbor, distance + 1));
                        }
                    }
                }
                if v[j] == k {
                    count += 1;
                }
            }
        }
        println!("{}", count);
    }
}