# Find first K characters in Nth term of the Thue-Morse sequence

• Last Updated : 23 Feb, 2022

Given two integers N and K, the task is to print the first K bits of the Nth term of the Thue-Morse sequence. The Thue-Morse sequence is a binary sequence. It starts with a “0” as its first term. And then after the next term is generated by replacing “0” with “01” and “1” with “10”.

Examples:

Input: N = 3, K = 2
Output: 01
Explanation: The 1st term is “0”.
The 2nd term is obtained by replacing “0” with “01” i.e. 2nd term is “01”.
The 3rd term in the sequence is obtained by replacing “0” with “01” and “1” with “10”.
So the 3rd term becomes “0110”. Hence, the 1st 2 characters of the 3rd term is “01”.

Input: N = 4, K = 7
Output: 0110100

Approach: The basic approach to solve this problem is to generate the Nth term of the sequence and print the first K characters of that term. This can be done using the algorithm discussed here.

Time Complexity: O(N * 2N)
Auxiliary Space: O(1)

Efficient Approach: The above approach can be optimized by observing that the ith term is the concatenation of (i – 1)th term and the inverse of (i – 1)th term where inverse means changing the polarity of all bits in a binary integer. Hence, xth term, Ai[x] = Ai-1[x – 1] if (x < 2i-1), otherwise Ai[x] = !Ai-1[x – 2i-1]. Therefore, using this relation, a recursive function can be created to calculate the value of each bit in the Nth term.

Below is the implementation of the above approach:

## C++

 `#include ``using` `namespace` `std;` `// Recursive function to find the``// value of the kth bit in Nth term``int` `findDig(``int` `N, ``long` `K,  ``int` `curr)``{``  ` `  ``// Base Case``  ``if` `(N == 0) {``    ``return` `curr;``  ``}` `  ``// Stores the middle index``  ``long` `middle = (``long``)``pow``(2, N) / 2;` `  ``// If K lies in 1st part``  ``if` `(K <= middle) {` `    ``// Recursive Call``    ``return` `findDig(N - 1, K, curr);``  ``}` `  ``// If K lies in 2nd part``  ``// having inverted value``  ``else` `{``    ``if` `(curr == 0) {``      ``curr = 1;``    ``}``    ``else` `{``      ``curr = 0;``    ``}` `    ``// Recursive Call``    ``return` `findDig(N - 1,``                   ``K - middle, curr);``  ``}``}` `// Function to find first K characters``// in Nth term of Thue-Morse sequence``void` `firstKTerms(``int` `N, ``int` `K)``{``  ` `  ``// Loop to iterate all K bits``  ``for` `(``int` `i = 1; i <= K; ++i)``  ``{` `    ``// Print value of ith bit``    ``cout << (findDig(N, i, 0));``  ``}``}` `// Driver Code``int` `main() {` `  ``int` `N = 4;``  ``int` `K = 7;` `  ``firstKTerms(N, K);``  ``return` `0;``}` `// This code is contributed by hrithikgarg03188.`

## Java

 `// Java Implementation of the above approach``import` `java.io.*;``import` `java.util.*;``class` `GFG {` `    ``// Recursive function to find the``    ``// value of the kth bit in Nth term``    ``public` `static` `int` `findDig(``int` `N, ``long` `K,``                              ``int` `curr)``    ``{``        ``// Base Case``        ``if` `(N == ``0``) {``            ``return` `curr;``        ``}` `        ``// Stores the middle index``        ``long` `middle = (``long``)Math.pow(``2``, N) / ``2``;` `        ``// If K lies in 1st part``        ``if` `(K <= middle) {` `            ``// Recursive Call``            ``return` `findDig(N - ``1``, K, curr);``        ``}` `        ``// If K lies in 2nd part``        ``// having inverted value``        ``else` `{``            ``if` `(curr == ``0``) {``                ``curr = ``1``;``            ``}``            ``else` `{``                ``curr = ``0``;``            ``}` `            ``// Recursive Call``            ``return` `findDig(N - ``1``,``                           ``K - middle, curr);``        ``}``    ``}` `    ``// Function to find first K characters``    ``// in Nth term of Thue-Morse sequence``    ``public` `static` `void` `firstKTerms(``int` `N, ``int` `K)``    ``{``        ``// Loop to iterate all K bits``        ``for` `(``int` `i = ``1``; i <= K; ++i) {` `            ``// Print value of ith bit``            ``System.out.print(findDig(N, i, ``0``));``        ``}``    ``}` `    ``// Driver Code``    ``public` `static` `void` `main(String args[])``    ``{``        ``int` `N = ``4``;``        ``int` `K = ``7``;` `        ``firstKTerms(N, K);``    ``}``}`

## Python3

 `# Recursive function to find the``# value of the kth bit in Nth term``def` `findDig(N, K, curr):` `    ``# Base Case``    ``if` `(N ``=``=` `0``):``        ``return` `curr` `    ``# Stores the middle index``    ``middle ``=` `pow``(``2``, N) ``/``/` `2` `    ``# If K lies in 1st part``    ``if` `(K <``=` `middle):` `        ``# Recursive Call``        ``return` `findDig(N ``-` `1``, K, curr)` `    ``# If K lies in 2nd part``    ``# having inverted value``    ``else``:``        ``if` `(curr ``=``=` `0``):``            ``curr ``=` `1` `        ``else``:``            ``curr ``=` `0` `        ``# Recursive Call``        ``return` `findDig(N ``-` `1``, K ``-` `middle, curr)` `# Function to find first K characters``# in Nth term of Thue-Morse sequence``def` `firstKTerms(N, K):` `    ``# Loop to iterate all K bits``    ``for` `i ``in` `range``(``1``, K``+``1``):` `     ``# Print value of ith bit``        ``print``(findDig(N, i, ``0``), end``=``"")` `# Driver Code``if` `__name__ ``=``=` `"__main__"``:` `    ``N ``=` `4``    ``K ``=` `7` `    ``firstKTerms(N, K)` `    ``# This code is contributed by rakeshsahni`

## C#

 `// C# Implementation of the above approach``using` `System;``class` `GFG``{` `  ``// Recursive function to find the``  ``// value of the kth bit in Nth term``  ``public` `static` `int` `findDig(``int` `N, ``long` `K, ``int` `curr)``  ``{``    ` `    ``// Base Case``    ``if` `(N == 0)``    ``{``      ``return` `curr;``    ``}` `    ``// Stores the middle index``    ``long` `middle = (``long``)Math.Pow(2, N) / 2;` `    ``// If K lies in 1st part``    ``if` `(K <= middle)``    ``{` `      ``// Recursive Call``      ``return` `findDig(N - 1, K, curr);``    ``}` `    ``// If K lies in 2nd part``    ``// having inverted value``    ``else``    ``{``      ``if` `(curr == 0)``      ``{``        ``curr = 1;``      ``}``      ``else``      ``{``        ``curr = 0;``      ``}` `      ``// Recursive Call``      ``return` `findDig(N - 1,``                     ``K - middle, curr);``    ``}``  ``}` `  ``// Function to find first K characters``  ``// in Nth term of Thue-Morse sequence``  ``public` `static` `void` `firstKTerms(``int` `N, ``int` `K)``  ``{``    ` `    ``// Loop to iterate all K bits``    ``for` `(``int` `i = 1; i <= K; ++i)``    ``{` `      ``// Print value of ith bit``      ``Console.Write(findDig(N, i, 0));``    ``}``  ``}` `  ``// Driver Code``  ``public` `static` `void` `Main()``  ``{``    ``int` `N = 4;``    ``int` `K = 7;` `    ``firstKTerms(N, K);``  ``}``}` `// This code is contributed by saurabh_jaiswal.`

## Javascript

 ``

Output
`0110100`

Time Complexity: O(K*log N)
Auxiliary Space: O(1)

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