**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 Java program to find the nth number in Calkin Wilf sequence:**

// 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; } }

**Output:**

[5, 3]

**Explanation:**

For n = 13,

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