description: You are given a matrix, consisting of n rows and m columns. The rows are numbered top to bottom, the columns are numbered left to right.
  
  Each cell of the matrix can be either free or locked.
  
  Let's call a path in the matrix a staircase if it: 
  
    * starts and ends in the free cell; 
    * visits only free cells; 
    * has one of the two following structures: 
      1. the second cell is 1 to the right from the first one, the third cell is 1 to the bottom from the second one, the fourth cell is 1 to the right from the third one, and so on; 
      2. the second cell is 1 to the bottom from the first one, the third cell is 1 to the right from the second one, the fourth cell is 1 to the bottom from the third one, and so on. 
  
  
  
  In particular, a path, consisting of a single cell, is considered to be a staircase.
  
  Here are some examples of staircases:
  
  <image>
  
  Initially all the cells of the matrix are free.
  
  You have to process q queries, each of them flips the state of a single cell. So, if a cell is currently free, it makes it locked, and if a cell is currently locked, it makes it free.
  
  Print the number of different staircases after each query. Two staircases are considered different if there exists such a cell that appears in one path and doesn't appear in the other path.
  
  Input
  
  The first line contains three integers n, m and q (1 ≤ n, m ≤ 1000; 1 ≤ q ≤ 10^4) — the sizes of the matrix and the number of queries.
  
  Each of the next q lines contains two integers x and y (1 ≤ x ≤ n; 1 ≤ y ≤ m) — the description of each query.
  
  Output
  
  Print q integers — the i-th value should be equal to the number of different staircases after i queries. Two staircases are considered different if there exists such a cell that appears in one path and doesn't appear in the other path.
  
  Examples
  
  Input
  
  
  2 2 8
  1 1
  1 1
  1 1
  2 2
  1 1
  1 2
  2 1
  1 1
  
  
  Output
  
  
  5
  10
  5
  2
  5
  3
  1
  0
  
  
  Input
  
  
  3 4 10
  1 4
  1 2
  2 3
  1 2
  2 3
  3 2
  1 3
  3 4
  1 3
  3 1
  
  
  Output
  
  
  49
  35
  24
  29
  49
  39
  31
  23
  29
  27
  
  
  Input
  
  
  1000 1000 2
  239 634
  239 634
  
  
  Output
  
  
  1332632508
  1333333000
solution: 
  import collections
  import math
  import itertools
  import  sys
  import time
  import random
  
  def fast_input():
      return sys.stdin.readline().strip()
  
  input = fast_input
  
  def ints_input():
      return list(int(x) for x in input().split())
  
  def int_input():
      return int(input())
  
  
  
  def solve():
      n, m, q = ints_input()
      free = [[1 for _ in range(m)] for _ in range(n)]
      staircases_on_diagonal = [0 for _ in range(-m, n)]
      total_non_singleton_staircases = 0
      total_singleton_staircases = n * m
      def calculate_staircases(x):
          if x <= 0:
              start_point = (0, -x)
          else:
              start_point = (x, 0)
          staircases = 0
          if start_point[0] > 0:
              running = free[start_point[0]-1][start_point[1]]
          else:
              running = 0
          while True:
              # print(x, start_point, running)
              # Update running
              if free[start_point[0]][start_point[1]]:
                  running += 1
              else:
                  running = 0
              if running > 1:
                  staircases += running - 1
              # Move right
              start_point = (start_point[0], start_point[1] + 1)
              if start_point[1] >= m:
                  break
              if free[start_point[0]][start_point[1]]:
                  running += 1
              else:
                  running = 0
              if running > 1:
                  staircases += running - 1
              # Move down
              start_point = (start_point[0] + 1, start_point[1])
              if start_point[0] >= n:
                  break
          return staircases
      for x in range(-(m-1), n):
          stairs = calculate_staircases(x)
          staircases_on_diagonal[x] = stairs
          total_non_singleton_staircases += stairs
      for _ in range(q):
          x, y = ints_input()
          x = x-1
          y = y-1
          free[x][y] = 1 - free[x][y]
          total_singleton_staircases += (1 if free[x][y] else -1)
          for diff in (x - y, x + 1 - y):
              if diff < -(m - 1) or diff >= n:
                  continue
              stairs = calculate_staircases(diff)
              total_non_singleton_staircases -= staircases_on_diagonal[diff]
              staircases_on_diagonal[diff] = stairs
              total_non_singleton_staircases += stairs
          print(total_non_singleton_staircases + total_singleton_staircases)
  
  
  
  
  
  
  if False:
      for _ in range(int_input()):
          solve()
  else:
      solve()