Baum Sweet Sequence

Baum Sweet Sequence is an infinite binary sequence of 0s and 1s. The nth term of the sequence is 1 if the number n has no odd number of contiguous zeroes in its binary representation, else the nth term is 0.

The first few terms of the sequence are:
b1 = 1 (binary of 1 is 1)
b2 = 0 (binary of 2 is 10)
b3 = 1 (binary of 3 is 11)
b4 = 1 (binary of 4 is 100)
b5 = 0 (binary of 5 is 101)
b6 = 0 (binary of 6 is 110)

Given a natural number n. The task is to find the nth term of the Baum Sweet sequence, i.e, check whether it contains any consecutive block of zeroes of odd length.

Input: n = 8
Output: 0
Explanations:
Binary representation of 8 is 1000. It 
contains odd length block of consecutive 0s. 
Therefore B8 is 0.

Input: n = 5
Output: 1

Input: n = 7
Output: 0

The idea is to run a loop through the binary representation of n and count the length of all the consecutive zero blocks present. If there is at-least one odd length zero block, then the nth term for the given input n is 0 else it is 1.

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// CPP code to find the nth term of the
// Baum Sweet Sequence
#include <bits/stdc++.h>
using namespace std;
  
int nthBaumSweetSeq(int n)
{
    // bitset stores bitwise representation
    bitset<32> bs(n);
  
    // len stores the number of bits in the 
    // binary of n. builtin_clz() function gives 
    // number of zeroes present before the 
    // leading 1 in binary of n
    int len = 32 - __builtin_clz(n);
  
    int baum = 1; // nth term of baum sequence
    for (int i = 0; i < len;) {
        int j = i + 1;
  
        // enter into a zero block
        if (bs[i] == 0) {
            int cnt = 1;
  
            // loop to run through each zero block
            // in binary representation of n
            for (j = i + 1; j < len; j++) {
  
                // counts consecutive zeroes 
                if (bs[j] == 0)                   
                    cnt++;
                else
                    break;
            }
  
            // check if the number of consecutive
            // zeroes is odd
            if (cnt % 2 == 1)
                baum = 0;
        }
        i = j;
    }
  
    return baum;
}
  
// Driver Code
int main()
{
    int n = 8;
    cout << nthBaumSweetSeq(n);
    return 0;
}

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Output:

0


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