Motzkin number

In mathematics, a Motzkin number for a given number n is the number of different ways of drawing non-intersecting chords between n points on a circle (not necessarily touching every point by a chord).
For example, for n = 3, M4 = 9.

Recurrence relation for Nth Motzkin Number is :

Motzkin Number can be used to find :

  • Number of positive integer sequences of length n – 1 in which the opening and ending elements are either 1 or 2, and the difference between any two consecutive elements is -1, 0 or 1.
  • Number of routes on the upper right quadrant of a grid from coordinate (0, 0) to coordinate (n, 0) in n steps if one is allowed to move only to the right (up, down or straight) at each step but forbidden from dipping below the y = 0 axis.
  • For example –

    Following figure shows the 9 valid Motzkin paths from (0, 0) to (4, 0).

    Examples :

    Input : n = 4
    Output : 9
    
    Input : n = 5
    Output : 21
    

    Below is the program to find nth Motzkin Number :

    C++

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    // CPP Program to find Nth Motzkin Number.
    #include <bits/stdc++.h>
    using namespace std;
      
    // Return the nth Motzkin Number.
    int motzkin(int n)
    {
        // Base Case
        if (n == 0 || n == 1)
            return 1;
      
        // Recursive step
        return ((2 * n + 1) * motzkin(n - 1) +
                (3 * n - 3) * motzkin(n - 2)) / (n + 2);
    }
      
    // Driven Program
    int main()
    {
        int n = 8;
        cout << motzkin(n) << endl;
        return 0;
    }

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    Java

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    // Java Program to find Nth Motzkin Number.
    import java.util.*;
      
    class Digits
    {
        // Return the nth Motzkin Number.
        public static int motzkin(int n)
        {
            // Base Case
            if (n == 0 || n == 1)
                return 1;
      
            // Recursive step
            return ((2 * n + 1) * motzkin(n - 1) +
                    (3 * n - 3) * motzkin(n - 2)) / (n + 2);
        
          
        // driver code    
        public static void main(String[] args)
        {
            int n = 8;
            System.out.print( motzkin(n) );
        }
    }
      
    // This code is contributed by rishabh_jain

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    Python3

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    # Python3 program to find Nth Motzkin Number.
      
    # Return the nth Motzkin Number.
    def motzkin(n) :
          
        # Base Case
        if (n == 0 or n == 1) :
            return 1
      
        # Recursive step
        return ((2 * n + 1) * motzkin(n - 1) + 
                (3 * n - 3) * motzkin(n - 2)) / (n + 2)
      
    # Driver code
    n = 8
    print( motzkin(n) )
      
    # This code is contributed by rishabh_jain

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    C#

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    // C# Program to find Nth Motzkin Number.
    using System;
      
    class GFG {
          
        // Return the nth Motzkin Number.
        public static int motzkin(int n)
        {
              
            // Base Case
            if (n == 0 || n == 1)
                return 1;
      
            // Recursive step
            return ((2 * n + 1) * motzkin(n - 1) +
                (3 * n - 3) * motzkin(n - 2)) / (n + 2);
        
          
        // driver code 
        public static void Main()
        {
            int n = 8;
              
            Console.WriteLine( motzkin(n) );
        }
    }
      
    // This code is contributed by vt_m

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    PHP

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    <?php
    // PHP Program to find
    // Nth Motzkin Number.
      
    // Return the nth Motzkin Number.
    function motzkin($n)
    {
        // Base Case
        if ($n == 0 || $n == 1)
            return 1;
      
        // Recursive step
        return ((2 * $n + 1) * 
                 motzkin($n - 1) +
                (3 * $n - 3) * 
                 motzkin($n - 2)) / 
                ($n + 2);
    }
      
    // Driven Code
    $n = 8;
    echo(motzkin($n));
      
    // This code is contributed by Ajit.
    ?>

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    Output :

    323
    



    Using Dynamic Programming :

    Below is the Dynamic Programming solution of finding nth Motzkin Number :

    C++

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    // CPP Program to find Nth Motzkin Number.
    #include <bits/stdc++.h>
    using namespace std;
      
    // Return the nth Motzkin Number.
    int motzkin(int n)
    {
        int dp[n + 1];
      
        // Base case
        dp[0] = dp[1] = 1;
      
        // Finding i-th Motzkin number.
        for (int i = 2; i <= n; i++)
            dp[i] = ((2 * i + 1) * dp[i - 1] + 
                      (3 * i - 3) * dp[i - 2]) / (i + 2);
      
        return dp[n];
    }
    // Driven Program
    int main()
    {
        int n = 8;
        cout << motzkin(n) << endl;
        return 0;
    }

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    Java

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    // Java Program to find Nth Motzkin Number.
    import java.util.*;
      
    class Digits
    {
        // Return the nth Motzkin Number.
        public static int motzkin(int n)
        {
            int[] dp = new int[n+1];
      
            // Base case
            dp[0] = dp[1] = 1;
      
            // Finding i-th Motzkin number.
            for (int i = 2; i <= n; i++)
                dp[i] = ((2 * i + 1) * dp[i - 1] + 
                    (3 * i - 3) * dp[i - 2]) / (i + 2);
      
            return dp[n];
        }
          
        // driver code    
        public static void main(String[] args)
        {
            int n = 8;
            System.out.print( motzkin(n) );
        }
    }
      
    // This code is contributed by rishabh_jain

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    Python3

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    # Python3 program to find Nth Motzkin Number.
      
    # Return the nth Motzkin Number.
    def motzkin(n) :
          
        dp = [None] * (n+1)
      
        # Base case
        dp[0] = dp[1] = 1;
      
        i = 2
        # Finding i-th Motzkin number.
        while i <= n :
            dp[i] = ((2 * i + 1) * dp[i - 1] + 
                    (3 * i - 3) * dp[i - 2]) / (i + 2);
            i = i + 1
        return dp[n];
      
    # Driver code
    n = 8
    print( motzkin(n) )
      
    # This code is contributed by rishabh_jain

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    C#

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    // C# Program to find Nth Motzkin Number.
    using System;
      
    class GFG {
          
        // Return the nth Motzkin Number.
        public static int motzkin(int n)
        {
            int[] dp = new int[n+1];
      
            // Base case
            dp[0] = dp[1] = 1;
      
            // Finding i-th Motzkin number.
            for (int i = 2; i <= n; i++)
                dp[i] = ((2 * i + 1) * dp[i - 1] + 
                 (3 * i - 3) * dp[i - 2]) / (i + 2);
      
            return dp[n];
        }
          
        // driver code 
        public static void Main()
        {
            int n = 8;
              
            Console.WriteLine( motzkin(n) );
        }
    }
      
    // This code is contributed by vt_m

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    PHP

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    <?php
    // PHP Program to find 
    // Nth Motzkin Number.
      
    // Return the nth Motzkin Number.
    function motzkin($n)
    {
      
        // Base case
        $dp[0] = $dp[1] = 1;
      
        // Finding i-th Motzkin number.
        for ($i = 2; $i <= $n; $i++)
            $dp[$i] = ((2 * $i + 1) * 
                        $dp[$i - 1] + 
                       (3 * $i - 3) * 
                       $dp[$i - 2]) / 
                       ($i + 2);
      
        return $dp[$n];
    }
    // Driven Code
    $n = 8;
    echo(motzkin($n));
      
    // This code is contributed by Ajit.
    ?>

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    Output :

    323
    


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