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 Fortran (gfortran v4.8.4) (judge verdict: Time Limit Exceeded)

program line_plus_plus
  implicit none
  integer :: n, x, y, i, j, k
  integer :: d(2001,2001) = 1e8, c(2000) = 0
  read(*,*) n, x, y
  do i = 1, n
    d(i,i) = 0
    d(i,i+1) = 1
    d(i+1,i) = 1
  end do
  d(x,y) = 1
  d(y,x) = 1
  do k = 1, n
    do i = 1, n
      do j = 1, n
        d(i,j) = min(d(i,j),d(i,k)+d(k,j))
        if (k == n) c(d(i,j)) = c(d(i,j))+1
      end do
    end do
  end do
  do k = 1, n-1
    write(*,'(i0)') c(k)/2
  end do
end program line_plus_plus