D - Line++

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.

5 2 4

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 $4$.

3 1 3

3
0

The graph in this input is as follows:

7 3 7

7
8
4
2
0
0

10 4 8

10
12
10
8
4
1
0
0
0