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We have a sequence generator that starts with 1. At each step 0 becomes 10 and 1 becomes 01. So following changes will occur at consecutive steps −

**Step 1** − 01

**Step 2** − 1001

**Step 3** − 01101001 ……

The goal is to find the number of pairs of consecutive 0’s for a given number of steps.

If input step is 1 pair of 0’s - 0, input step is 2 pair of 0’s - 1, input step is 3 pair of 0’s 1

**Step 4** − 1001011001101001

**Step 5** − 01101001100101101001011001101001

We can observe that sequence is increasing in powers of 2 and repeats itself after length 12 and for every 12 characters. So, these 12 length sequences have 2 pairs of consecutive 0’s.

For steps S number of 12 length patterns will be 2S /12

Number of consecutive 2 patterns = 1 (initial) + 2 X S ( for rest of 12 length repetitions )

Let us understand with examples.

**Input** − steps = 5

**Output** − Count of Pairs Of Consecutive Zeros are − 5

**Explanation** − As shown above sequence at 5th step is −

Step 5: 01101001100101101001011001101001 Number of pairs of 0’s is 5. Also with formula : tmp=25 /12= 32/12 = 2, pairs=1+ 2 x 2 = 5

**Input** − steps = 10

**Output** − Count of Pairs Of Consecutive Zeros are − 171

**Explanation** − With formula − tmp=210/12= 1024/12 = 85, pairs=1+ 2 x 85 = 171

We will take the number of steps as input and check if steps=1 then the number of pairs of consecutive 0’s is 0. If steps = 2 or 3 then the number of such pairs is 1.

Otherwise, calculate using formula tmp=2input/12 and pairs=1+2*tmp.

Take a variable decimal as input for steps.

Function Zero_pairs(int decimal) takes decimal and returns Count of Pairs Of Consecutive Zeros for that step.

Take the initial count as 0.

If decimal<=1, return 0.

If decimal==2 or decimal==3. Then return 1.

Else calculate temp=(pow(2,decimal))/12 and calculate count as 2*temp + 1.

Return count as result.

#include<bits/stdc++.h> using namespace std; int Zero_pairs(int decimal){ int count = 0; if(decimal <=1){ count = 0; } else if(decimal == 2 || decimal == 3){ count = 1; } else{ int temp = (pow(2, decimal) / 12); count = 2 * temp + 1; } return count; } int main(){ int decimal = 7; cout<<"Count of Pairs Of Consecutive Zeros are: "<<Zero_pairs(decimal); return 0; }

If we run the above code it will generate the following output −

Count of Pairs Of Consecutive Zeros are: 21

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