nth Rational number in Calkin-Wilf sequence

What is Calkin Wilf Sequence?
A Calkin-Wilf tree (or sequence) is a special binary tree which is obtained by starting with the fraction 1/1 and adding a/(a+b) and (a+b)/b iteratively below each fraction a/b. This tree generates every rational number. Writing out the terms in a sequence gives 1/1, 1/2, 2/1, 1/3, 3/2, 2/3, 3/1, 1/4, 4/3, 3/5, 5/2, 2/5, 5/3, 3/4, 4/1, …The sequence has the property that each denominator is the next numerator.

The image above is the Calkin-Wilf Tree where all the rational numbers are listed. The children of a node a/b is calculated as a/(a+b) and (a+b)/b.

The task is to find the nth rational number in breadth first traversal of this tree.
Examples:

Input : 13
Output : [5, 3]

Input : 5
Output : [3, 2]

Explanation: This tree is a Perfect Binary Search tree and we need floor(log(n)) steps to compute nth rational number. The concept is similar to searching in a binary search tree. Given n we keep dividing it by 2 until we get 0. We return fraction at each stage in the following manner:-



    if n%2 == 0
        update frac[1]+=frac[0]
    else
        update frac[0]+=frac[1]



Below is the program to find the nth number in Calkin Wilf sequence:

C++

// C++ program to find the 
// nth number in Calkin
// Wilf sequence:
# include<bits/stdc++.h>
using namespace std;
  
int frac[] = {0, 1};
  
// returns 1x2 int array  
// which contains the nth
// rational number
int nthRational(int n)
{
    if (n > 0)
        nthRational(n / 2);
  
    // ~n&1 is equivalent to 
    // !n%2?1:0 and n&1 is 
    // equivalent to n%2
    frac[~n & 1] += frac[n & 1];
}
  
// Driver Code
int main()
{
    int n = 13; // testing for n=13
      
    // converting array 
    // to string format
    nthRational(n);
    cout << "[" << frac[0] << "," 
         << frac[1] << "]" << endl;
    return 0;
}
  
// This code is contributed
// by Harshit Saini

Java

// Java program to find the nth number
// in Calkin Wilf sequence:
import java.util.*;
  
public class GFG {
    static int[] frac = { 0, 1 };
  
    public static void main(String args[])
    {
        int n = 13; // testing for n=13
  
        // converting array to string format
        System.out.println(Arrays.toString(nthRational(n)));
    }
  
    // returns 1x2 int array which 
    // contains the nth rational number
    static int[] nthRational(int n)
    {
        if (n > 0)
            nthRational(n / 2);
  
        // ~n&1 is equivalent to !n%2?1:0 
        // and n&1 is equivalent to n%2
        frac[~n & 1] += frac[n & 1];
          
  
        return frac;
    }
}

Python3

# Python program to find 
# the nth number in Calkin
# Wilf sequence:
frac = [0, 1]
  
# returns 1x2 int array 
# which contains the nth
# rational number
def nthRational(n):
    if n > 0:
        nthRational(int(n / 2))
      
    # ~n&1 is equivalent to 
    # !n%2?1:0 and n&1 is 
    # equivalent to n%2
    frac[~n & 1] += frac[n & 1]
      
    return frac
  
# Driver code
if __name__ == "__main__":
      
    n = 13 # testing for n=13
      
    # converting array 
    # to string format
    print(nthRational(n))
      
# This code is contributed
# by Harshit Saini

Output:

[5, 3]

Explanation:
For n = 13,



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Improved By : Harshit Saini




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