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 PyPy3 (2.4.0) (judge verdict: Accepted)

import heapq

def djsktra(node,edges,dist):
    heap = []
    heapq.heappush(heap,(0,node))
    visited = set()
    while heap:
        curr = heapq.heappop(heap)
        d,node = curr[0],curr[1]
        if node not in visited:
            visited.add(node)
            dist[d] += 1
            for kid in edges[node]:
                heapq.heappush(heap,(d+1,kid))

def main():
    n,x,y = map(int,input().split())
    edges = {}
    for i in range(1,n):
        if i not in edges.keys():
            edges[i] = []

        if i+1 not in edges.keys():
            edges[i+1] = []
            
        edges[i].append(i+1)
        edges[i+1].append(i)
        
    edges[x].append(y)
    edges[y].append(x)
    dist = [0]*n
    for i in range(1,n+1):
        djsktra(i,edges,dist)

    dist.pop(0)
    for i in dist:
        print(i//2)

main()