Consider a sequence that starts with a 1 on a machine. At each successive step, the machine simultaneously transforms each digit 0 into the sequence 10 and each digit 1 into the sequence 01.

After the first time step, the sequence 01 is obtained; after the second, the sequence 1001, after the third, the sequence 01101001 and so on.

How many pairs of consecutive zeros will appear in the sequence after n steps?

Examples:

Input : Number of steps = 3 Output: 1 // After 3rd step sequence will be 01101001 Input : Number of steps = 4 Output: 3 // After 4rd step sequence will be 1001011001101001 Input : Number of steps = 5 Output: 5 // After 3rd step sequence will be 01101001100101101001011001101001

This is a simple reasoning problem. If we see the sequence very carefully , then we will be able to find a pattern for given sequence. If n=1 sequence will be {01} so number of pairs of consecutive zeros are 0, If n = 2 sequence will be {1001} so number of pairs of consecutive zeros are 1, If n=3 sequence will be {01101001} so number of pairs of consecutive zeros are 1,

If n=4 sequence will be {1001011001101001} so number of pairs of consecutive zeros are 3.

So length of the sequence will always be a power of 2. We can see after length 12 sequence is repeating and in lengths of 12. And in a segment of length 12, there are total 2 pairs of consecutive zeros. Hence we can generalize the given pattern q = (2^n/12) and total pairs of consecutive zeros will be 2*q+1.

// C++ program to find number of consecutive // 0s in a sequence #include<bits/stdc++.h> using namespace std; // Function to find number of consecutive Zero Pairs // Here n is number of steps int consecutiveZeroPairs(int n) { // Base cases if (n==1) return 0; if (n==2 || n==3) return 1; // Calculating how many times divisible by 12, i.e., // count total number repeating segments of length 12 int q = (pow(2,n) / 12); // number of consecutive Zero Pairs return 2*q + 1; } // Driver program to run the test case int main() { int n = 5; cout << consecutiveZeroPairs(n) << endl; return 0; }

Output:

5

This article is contributed by **Shashank Mishra ( Gullu )**. this article is reviewed by team GeeksForGeeks.

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