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Thue-Morse sequence
  • Difficulty Level : Easy
  • Last Updated : 05 Apr, 2021

Thue–Morse sequence, or Prouhet–Thue–Morse sequence, is an infinite binary sequence of 0s and 1s. The sequence is obtained by starting with 0 and successively appending the Boolean complement of the sequence obtained so far.
First few steps : 

Start with 0 
Append complement of 0, we get 01 
Append complement of 01, we get 0110 
Append complement of 0110, we get 01101001

Given a whole number n. The task is to find the nth string formed of by Thue–Morse sequence i.e prefix of length 2n-1 of Thue–Morse sequence. 
Examples: 
 

Input : n = 4
Output : 01101001
We get 0, 01, 0110 and 01101001
in fourth iteration.

Input : n = 3
Output : 0110

The idea is to initialize the output string with 0, then run a loop n – 1 times and for each iteration find the complement of the string and append it to the string.
Below is implementation of this approach: 
 

C++




// CPP Program to find nth term of Thue-Morse sequence.
#include <bits/stdc++.h>
using namespace std;
 
// Return the complement of the binary string.
string complement(string s)
{
    string comps;
 
    // finding the complement of the string.
    for (int i = 0; i < s.length(); i++) {
 
        // if character is 0, append 1
        if (s.at(i) == '0')
            comps += '1';
 
        // if character is 1, append 0.
        else
            comps += '0';
    }
 
    return comps;
}
 
// Return the nth term of Thue-Morse sequence.
string nthTerm(int n)
{
    // Initialing the string to 0
    string s = "0";
 
    // Running the loop for n - 1 time.
    for (int i = 1; i < n; i++)
 
        // appending the complement of
        // the string to the string.
        s += complement(s);
     
 
    return s;
}
// Driven Program
int main()
{
    int n = 4;
    cout << nthTerm(n) << endl;
    return 0;
}

Java




// Java Program to find nth
// term of Thue-Morse sequence.
 
class GFG
{
    // Return the complement
    // of the binary String.
    static String complement(String s)
    {
        String comps = "";
     
        // finding the complement
        // of the String.
        for (int i = 0; i < s.length(); i++)
        {
     
            // if character is 0,
            // append 1
            if (s.charAt(i) == '0')
                comps += '1';
     
            // if character is 1,
            // append 0.
            else
                comps += '0';
        }
     
        return comps;
    }
     
    // Return the nth term
    // of Thue-Morse sequence.
    static String nthTerm(int n)
    {
        // Initialing the
        // String to 0
        String s = "0";
     
        // Running the loop
        // for n - 1 time.
        for (int i = 1; i < n; i++)
     
            // appending the complement of
            // the String to the String.
            s += complement(s);
         
     
        return s;
    }
     
    // Driven Code
public static void main(String[] args)
    {
        int n = 4;
        System.out.print(nthTerm(n));
    }
}
 
// This code is contributed by
// mits

Python3




# Python3 Program to find nth term of
# Thue-Morse sequence.
 
# Return the complement of the
# binary string.
def complement(s):
    comps = "";
 
    # finding the complement
    # of the string.
    for i in range(len(s)):
 
        # if character is 0, append 1
        if (s[i] == '0'):
            comps += '1';
 
        # if character is 1, append 0.
        else:
            comps += '0';
 
    return comps;
 
# Return the nth term of
# Thue-Morse sequence.
def nthTerm(n):
 
    # Initialing the string to 0
    s = "0";
 
    # Running the loop for n - 1 time.
    for i in range(1, n):
 
        # appending the complement of
        # the string to the string.
        s += complement(s);
     
    return s;
 
# Driver Code
n = 4;
print(nthTerm(n));
 
# This code is contributed
# by mits

C#




// C# Program to find nth
// term of Thue-Morse sequence.
using System;
 
class GFG
{
    // Return the complement
    // of the binary string.
    static string complement(string s)
    {
        string comps = "";
     
        // finding the complement
        // of the string.
        for (int i = 0; i < s.Length; i++)
        {
     
            // if character is 0,
            // append 1
            if (s[i] == '0')
                comps += '1';
     
            // if character is 1,
            // append 0.
            else
                comps += '0';
        }
     
        return comps;
    }
     
    // Return the nth term
    // of Thue-Morse sequence.
    static string nthTerm(int n)
    {
        // Initialing the
        // string to 0
        string s = "0";
     
        // Running the loop
        // for n - 1 time.
        for (int i = 1; i < n; i++)
     
            // appending the complement of
            // the string to the string.
            s += complement(s);
         
     
        return s;
    }
     
    // Driven Code
    static void Main()
    {
        int n = 4;
        Console.Write(nthTerm(n));
    }
}
 
// This code is contributed by
// Manish Shaw(manishshaw1)

PHP




<?php
// PHP Program to find nth
// term of Thue-Morse sequence.
 
// Return the complement
// of the binary string.
function complement($s)
{
    $comps = "";
 
    // finding the complement
    // of the string.
    for ($i = 0;
         $i < strlen($s); $i++)
    {
 
        // if character is
        // 0, append 1
        if ($s[$i] == '0')
            $comps .= '1';
 
        // if character is
        // 1, append 0.
        else
            $comps .= '0';
    }
 
    return $comps;
}
 
// Return the nth term
// of Thue-Morse sequence.
function nthTerm($n)
{
    // Initialing the
    // string to 0
    $s = "0";
 
    // Running the loop
    // for n - 1 time.
    for ($i = 1; $i < $n; $i++)
 
        // appending the complement of
        // the string to the string.
        $s .= complement($s);
     
 
    return $s;
}
 
// Driven Code
$n = 4;
echo nthTerm($n);
 
// This code is contributed
// by mits
?>

C++




#include <iostream>
using namespace std;
 
int main() {
 
    cout<<"GFG!";
    return 0;
}

Javascript




<script>
 
// JavaScript Program to find nth
// term of Thue-Morse sequence.
 
    // Return the complement
    // of the binary string.
    function complement(s)
    {
        let comps = "";
       
        // finding the complement
        // of the string.
        for (let i = 0; i < s.length; i++)
        {
       
            // if character is 0,
            // append 1
            if (s[i] == '0')
                comps += '1';
       
            // if character is 1,
            // append 0.
            else
                comps += '0';
        }
       
        return comps;
    }
       
    // Return the nth term
    // of Thue-Morse sequence.
    function nthTerm(n)
    {
        // Initialing the
        // string to 0
        let s = "0";
       
        // Running the loop
        // for n - 1 time.
        for (let i = 1; i < n; i++)
       
            // appending the complement of
            // the string to the string.
            s += complement(s);
           
       
        return s;
    }
 
// Driver program
 
        let n = 4;
        document.write(nthTerm(n));
         
        // This code is contributed by susmitakundugoaldanga.
</script>

Output: 
 

01101001

 

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