Thue-Morse sequence

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 2^n-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)
{
    // Initializing 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)
    {
        // Initializing 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):

    # Initializing 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)
    {
        // Initializing 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)
{
    // Initializing 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
?>

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)
    {
        // Initializing 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

Time Complexity: O(n*log(n))
Auxiliary Complexity: O(1)