Count Pairs Of Consecutive Zeros

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.
Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above.

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