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 to the Nth Motzkin Number is:
Motzkin Number can be used to find:
- The 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.
- The 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–
The 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++
// 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 Programint main(){ int n = 8; cout << motzkin(n) << endl; return 0;} |
Java
// 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 |
Python3
# 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 coden = 8print( motzkin(n) ) # This code is contributed by rishabh_jain |
C#
// 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 |
PHP
<?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.?> |
Javascript
<script>// javascript 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); } // driver code var n = 8; document.write( motzkin(n) );</script> |
Output :
323
Time complexity: O(2n)
space complexity: O(2n)
Using Dynamic Programming :
Below is the Dynamic Programming solution of finding nth Motzkin Number :
C++
// 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 Programint main(){ int n = 8; cout << motzkin(n) << endl; return 0;} |
Java
// 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 |
Python3
# 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 coden = 8print( motzkin(n) ) # This code is contributed by rishabh_jain |
C#
// 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 |
PHP
<?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.?> |
Javascript
// JavaScript Program to find Nth Motzkin Number.// Return the nth Motzkin Number.function motzkin(n){ let dp = new Array(n+1).fill(0); // Base case dp[0] = dp[1] = 1; // Finding i-th Motzkin number. for (let 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 Programlet n = 8;console.log(motzkin(n));// The code is contributed by Nidhi goel |
Output:
323
Time complexity: O(n)
space complexity: O(n)
Feeling lost in the world of random DSA topics, wasting time without progress? It’s time for a change! Join our DSA course, where we’ll guide you on an exciting journey to master DSA efficiently and on schedule.
Ready to dive in? Explore our Free Demo Content and join our DSA course, trusted by over 100,000 neveropen!
Ready to dive in? Explore our Free Demo Content and join our DSA course, trusted by over 100,000 neveropen!
