Motzkin number in C++

The Motzkin number series starts with 1, 1, 4, 9, etc.., We can get the generalised nth term with the sequence. The Motzkin number sequence is as follows.

a₀ = 1

a₁ = 1

a₂ = 4

a₃ = 9

a_n = ((2 * n + 1)/ n + 2) * M_(n-1) +((3 * n - 3)/ n + 2) * M_{(n - 2)}

Algorithm

• Initialise the number n.

• Iterate till n.

  • Update the previous two numbers

• Return the last number.

Example

Implementation

Following is the implementation of the above algorithm in C++

#include <bits/stdc++.h>
using namespace std;
int getNthTerm(int n) {
   if(n == 0 || n == 1) {
      return 1;
   }
   int a = 1, b = 1;
   for(int i = 2; i <= n; ++i) {
      int c = ((2 * i + 1) * b + (3 * i - 3) * a) / (i + 2);
      a = b;
      b = c;
   }
   return b;
}
int main() {
   int n = 5;
   cout << getNthTerm(n) << endl;
   return 0;
}

Output

If you run the above code, then you will get the following result.

21