Problem: There are some coins in the xy-plane. The positions of the coins are represented by a grid of characters with H rows and W columns. If the character at the i-th row and j-th column, s_{ij}, is `#`, there is one coin at point (i,j); if that character is `.`, there is no coin at point (i,j). There are no other coins in the xy-plane.

There is no coin at point (x,y) where 1\leq i\leq H,1\leq j\leq W does not hold. There is also no coin at point (x,y) where x or y (or both) is not an integer. Additionally, two or more coins never exist at the same point.

Find the number of triples of different coins that satisfy the following condition:

* Choosing any two of the three coins would result in the same Manhattan distance between the points where they exist.



Here, the Manhattan distance between points (x,y) and (x',y') is |x-x'|+|y-y'|. Two triples are considered the same if the only difference between them is the order of the coins.

Constraints

* 1 \leq H,W \leq 300
* s_{ij} is `#` or `.`.

Input

Input is given from Standard Input in the following format:


H W
s_{11}...s_{1W}
:
s_{H1}...s_{HW}


Output

Print the number of triples that satisfy the condition.

Examples

Input

5 4
#.##
.##.
#...
..##
...#


Output

3


Input

5 4
.##
.##.
...
..##
...#


Output

3


Input

13 27
......#.........#.......#..
...#.....###..
..............#####...##...
...#######......#...#######
...#.....#.....###...#...#.
...#######....#.#.#.#.###.#
..............#.#.#...#.#..
.#.#.#...###..
...........#...#...#######
..#######..#...#...#.....#
..#.....#..#...#...#.###.#
..#######..#...#...#.#.#.#
..........##...#...#.#####


Output

870

Solution (in JAVA):

Solution:

import java.util.ArrayList;
import java.util.List;

public class Main {


  private static void solve() {
    int h = ni();
    int w = ni();

    int nh = h + w;
    int nw = h + w;

    List<int[]> list = new ArrayList<>();
    int[][] map = new int[nh][nw];
    for (int i = 0; i < h; i++) {
      char[] s = ns();
      for (int j = 0; j < w; j++) {
        if (s[j] == '#') {
          int ny = i + j;
          int nx = w - 1 + i - j;
          list.add(new int[] {ny, nx});
          map[ny][nx] = 1;
        }
      }
    }
    
    int[][] sumH = new int[nh][nw + 1];
    for (int i = 0; i < nh; i ++) {
      for (int j = 0; j < nw; j ++) {
        sumH[i][j + 1] = sumH[i][j] + map[i][j];
      }
    }
    
    int[][] sumV = new int[nh + 1][nw];
    for (int j = 0; j < nw; j ++) {
      for (int i = 0; i < nh; i ++) {
        sumV[i + 1][j] = sumV[i][j] + map[i][j];
      }
    }
    
    long ret = 0;
    for (int i = 0; i < nh; i ++) {
      for (int j = 0; j < nw; j ++) {
        if (map[i][j] == 0) continue;
        for (int k = j + 1; k < nw; k ++) {
          if (map[i][k] == 0) continue;
          int d = k - j;
          
          
          if (i + d < nh) ret += sumH[i + d][k + 1] - sumH[i + d][j];
          if (i - d >= 0) ret += sumH[i - d][k + 1] - sumH[i - d][j];
        }
      }
    }

    for (int i = 0; i < nw; i ++) {
      for (int j = 0; j < nh; j ++) {
        if (map[j][i] == 0) continue;
        for (int k = j + 2; k < nh; k ++) {
          if (map[k][i] == 0) continue;
          int d = k - j;
          
          
          if (i + d < nw) ret += sumV[k][i + d] - sumV[j + 1][i + d];
          if (i - d >= 0) ret += sumV[k][i - d] - sumV[j + 1][i - d];
        }
      }
    }

    System.out.println(ret);
  }

  public static void main(String[] args) {
    new Thread(null, new Runnable() {
      @Override
      public void run() {
        long start = System.currentTimeMillis();
        String debug = args.length > 0 ? args[0] : null;
        if (debug != null) {
          try {
            is = java.nio.file.Files.newInputStream(java.nio.file.Paths.get(debug));
          } catch (Exception e) {
            throw new RuntimeException(e);
          }
        }
        reader = new java.io.BufferedReader(new java.io.InputStreamReader(is), 32768);
        solve();
        out.flush();
        tr((System.currentTimeMillis() - start) + "ms");
      }
    }, "", 64000000).start();
  }

  private static java.io.InputStream is = System.in;
  private static java.io.PrintWriter out = new java.io.PrintWriter(System.out);
  private static java.util.StringTokenizer tokenizer = null;
  private static java.io.BufferedReader reader;

  public static String next() {
    while (tokenizer == null || !tokenizer.hasMoreTokens()) {
      try {
        tokenizer = new java.util.StringTokenizer(reader.readLine());
      } catch (Exception e) {
        throw new RuntimeException(e);
      }
    }
    return tokenizer.nextToken();
  }

  private static double nd() {
    return Double.parseDouble(next());
  }

  private static long nl() {
    return Long.parseLong(next());
  }

  private static int[] na(int n) {
    int[] a = new int[n];
    for (int i = 0; i < n; i++)
      a[i] = ni();
    return a;
  }

  private static char[] ns() {
    return next().toCharArray();
  }

  private static long[] nal(int n) {
    long[] a = new long[n];
    for (int i = 0; i < n; i++)
      a[i] = nl();
    return a;
  }

  private static int[][] ntable(int n, int m) {
    int[][] table = new int[n][m];
    for (int i = 0; i < n; i++) {
      for (int j = 0; j < m; j++) {
        table[i][j] = ni();
      }
    }
    return table;
  }

  private static int[][] nlist(int n, int m) {
    int[][] table = new int[m][n];
    for (int i = 0; i < n; i++) {
      for (int j = 0; j < m; j++) {
        table[j][i] = ni();
      }
    }
    return table;
  }

  private static int ni() {
    return Integer.parseInt(next());
  }

  private static void tr(Object... o) {
    if (is != System.in)
      System.out.println(java.util.Arrays.deepToString(o));
  }
}