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 Python (3.8.2) (judge verdict: Accepted)


def search(start, N, X, Y):
    dist = [0 for _ in range(N)]
    if start <= X:
        for i in range(X):
            dist[i] = abs(start - i)
        for i in range(Y, N):
            dist[i] = (X - start) + 1 + (i - Y)
        for i in range(X, (Y - X) // 2 + X + 1):
            dist[i] = (i - start)
        for i in range((Y - X) // 2 + X + 1, Y):
            dist[i] = (X - start) + 1 + (Y - i)
    elif start >= Y:
        for i in range(Y, N):
            dist[i] = abs(start - i)
        for i in range(X):
            dist[i] = (start - Y) + 1 + (X - i)
        for i in range(X, (Y - X) // 2 + X):
            dist[i] = (start - Y) + 1 + (i - X)
        for i in range((Y - X) // 2 + X, Y):
            dist[i] = (start - i)
    else:
        toX = min(start - X, Y - start + 1)
        toY = min(Y - start, start - X + 1)
        dist[start] = 0
        for i in range(X):
            dist[i] = (X - i) + toX
        for i in range(Y, N):
            dist[i] = toY + (i - Y)
        for i in range(X, start):
            dist[i] = min(start - i, Y - start + 1 + i - X)
        for i in range(start + 1, Y):
            dist[i] = min(i - start, start - X + 1 + Y - i)
    return dist


def main():
    N, X, Y = [int(n) for n in input().split(" ")]
    #N, X, Y = 10, 3, 8

    X = X - 1
    Y = Y - 1

    lenD = []
    for i in range(N):
        d = search(i, N, X, Y)
        lenD.append(d)

    kcounter = [0 for _ in range(N - 1)]

    for i in range(N):
        for j in range(i + 1, N):
            k = lenD[i][j]
            kcounter[k - 1] += 1

    for k in kcounter:
        print(k)

main()