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Find nth term of the Dragon Curve Sequence

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  • Difficulty Level : Easy
  • Last Updated : 19 Sep, 2022
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Dragon Curve Sequence is an infinite binary sequence of 0’s and 1’s. The first term of the sequence is 1. 

From the next term, we alternately insert 1 and 0 between each element of the previous term. 
To understand better refer the following explanations:

  • 1 (starts with 1) 
     
  • “1” 1 “0” 
    1 and 0 are inserted alternately to the left and right of the previous term. Here the number in the double quotes represents the newly added elements.
    So the second term becomes 
    1 1 0
  • “1” 1 “0” 1 “1” 0 “0” 
    So the third term becomes 
    1 1 0 1 1 0 0 
     
  • “1” 1 “0” 1 “1” 0 “0” 1 “1” 1 “0” 0 “1” 0 “0” 
    The fourth term becomes 
    1 1 0 1 1 0 0 1 1 1 0 0 1 0 0 
     

This is also popularly known as the regular paperfolding sequence. Given a natural number n, the task is to find the nth string formed by Dragon Curve sequence of length 2^n - 1          .

Examples: 

Input: n = 4
Output: 110110011100100
Explanation:
We get 1 as the first term, 
"110" as the second term,
"1101100" as the third term ,
And hence our fourth term will be
"110110011100100"

Input: n = 3
Output: 1101100

Approach: 

  • Step 1: Start with the first term 1. Then add 1 and 0 alternately after each element of the preceding term. 
  • Step 2: The new term obtained becomes the current term.
  • Step 3: Repeat the process in a loop from 1 to n, to generate each term and finally the nth term.

Algorithm :

  • Step 1: Take the input size n 
  • Step 2: Initialize 1st term of string as “1”.
  • step 3: generate each term of the sequence using nested for loop.
  • Step 4: Add alternate 0 and 1 in between, if previous term is 1 then add 0; vice versa. 
  • Step 5: Print the output string.

Below is the implementation of above idea:

C++




// CPP code to find nth term
// of the Dragon Curve Sequence
#include <bits/stdc++.h>
using namespace std;
 
// function to generate the nth term
string Dragon_Curve_Sequence(int n)
{
    // first term
    string s = "1";
 
    // generating each term of the sequence
    for (int i = 2; i <= n; i++)
    {
        string temp = "1";
        char prev = '1', zero = '0', one = '1';
 
        // loop to generate the ith term
        for (int j = 0; j < s.length(); j++)
        {
            // add character from the
            // original string
            temp += s[j];
 
            // add alternate 0 and 1 in between
            if (prev == '0')
            {
                // if previous added term
                // was '0' then add '1'
                temp += one;
 
                // now current term becomes
                // previous term
                prev = one;
            }
            else
            {
                // if previous added term
                // was '1', then add '0'
                temp += zero;
 
                // now current term becomes
                // previous term
                prev = zero;
            }
        }
         
        // s becomes the ith term of the sequence
        s = temp;
    }
    return s;
}
 
// Driver program
int main()
{
    // Taking inputs
    int n = 4;
 
    // generate nth term of dragon curve sequence
    string s = Dragon_Curve_Sequence(n);
     
    // Printing output
    cout << s << "\n";
}

Java




// Java code to find nth term
// of the Dragon Curve Sequence
import java.util.*;
 
class solution
{
 
// function to generate the nth term
static String Dragon_Curve_Sequence(int n)
{
    // first term
    String s = "1";
 
    // generating each term of the sequence
    for (int i = 2; i <= n; i++)
    {
        String temp = "1";
        char prev = '1', zero = '0', one = '1';
 
        // loop to generate the ith term
        for (int j = 0; j < s.length(); j++)
        {
            // add character from the
            // original string
            temp += s.charAt(j);
 
            // add alternate 0 and 1 in between
            if (prev == '0')
            {
                // if previous added term
                // was '0' then add '1'
                temp += one;
 
                // now current term becomes
                // previous term
                prev = one;
            }
            else
            {
                // if previous added term
                // was '1', then add '0'
                temp += zero;
 
                // now current term becomes
                // previous term
                prev = zero;
            }
        }
         
        // s becomes the ith term of the sequence
        s = temp;
    }
    return s;
}
 
// Driver program
public static void main(String args[])
{
    // Taking inputs
    int n = 4;
 
    // generate nth term of dragon curve sequence
    String s = Dragon_Curve_Sequence(n);
     
    // Printing output
    System.out.println(s);
}
 
}
 
//This code is contributed by
//Surendra_Gangwar

Python




# Python code to find nth term
# of the Dragon Curve Sequence
 
# function to generate
# the nth term
def Dragon_Curve_Sequence(n):
     
    # first term
    s = "1"
 
    # generating each term
    # of the sequence
    for i in range(2, n + 1):
        temp = "1"
        prev = '1'
        zero = '0'
        one = '1'
 
        # loop to generate the ith term
        for j in range(len(s)):
             
            # add character from the
            # original string
            temp += s[j]
 
            # add alternate 0 and
            # 1 in between
            if (prev == '0'):
                 
                # if previous added term
                # was '0' then add '1'
                temp += one
 
                # now current term becomes
                # previous term
                prev = one
 
            else:
                 
                # if previous added term
                # was '1', then add '0'
                temp += zero
 
                # now current term becomes
                # previous term
                prev = zero
 
        # s becomes the ith term
        # of the sequence
        s = temp
 
    return s
 
# Driver Code
n = 4
 
# generate nth term of
# dragon curve sequence
s = Dragon_Curve_Sequence(n)
 
# Printing output
print(s)
 
# This code is contributed by
# Sanjit_Prasad

C#




// C# code to find nth term
// of the Dragon Curve Sequence
using System;
 
class GFG
{
 
// function to generate the nth term
static String Dragon_Curve_Sequence(int n)
{
    // first term
    String s = "1";
 
    // generating each term of the sequence
    for (int i = 2; i <= n; i++)
    {
        String temp = "1";
        char prev = '1', zero = '0', one = '1';
 
        // loop to generate the ith term
        for (int j = 0; j < s.Length; j++)
        {
            // add character from the
            // original string
            temp += s[j];
 
            // add alternate 0 and 1 in between
            if (prev == '0')
            {
                // if previous added term
                // was '0' then add '1'
                temp += one;
 
                // now current term becomes
                // previous term
                prev = one;
            }
            else
            {
                // if previous added term
                // was '1', then add '0'
                temp += zero;
 
                // now current term becomes
                // previous term
                prev = zero;
            }
        }
 
        // s becomes the ith term of the sequence
        s = temp;
    }
    return s;
}
 
// Driver Code
public static void Main()
{
    // Taking inputs
    int n = 4;
 
    // generate nth term of dragon
    // curve sequence
    String s = Dragon_Curve_Sequence(n);
 
    // Printing output
    Console.WriteLine(s);
}
}
 
// This code is contributed by Rajput-Ji

PHP




<?php
// PHP code to find nth term
// of the Dragon Curve Sequence
 
// function to generate the nth term
function Dragon_Curve_Sequence($n)
{
    // first term
    $s = "1";
 
    // generating each term of the sequence
    for ($i = 2; $i <= $n; $i++)
    {
        $temp = "1";
        $prev = '1';
        $zero = '0';
        $one = '1';
 
        // loop to generate the ith term
        for ($j = 0; $j < strlen($s); $j++)
        {
            // add character from the
            // original string
            $temp .= $s[$j];
 
            // add alternate 0 and 1 in between
            if ($prev == '0')
            {
                // if previous added term
                // was '0' then add '1'
                $temp .= $one;
 
                // now current term becomes
                // previous term
                $prev = $one;
            }
            else
            {
                // if previous added term
                // was '1', then add '0'
                $temp .= $zero;
 
                // now current term becomes
                // previous term
                $prev = $zero;
            }
        }
         
        // s becomes the ith term of the sequence
        $s = $temp;
    }
    return $s;
}
 
// Driver code
 
    // Taking inputs
    $n = 4;
 
    // generate nth term of dragon curve sequence
    $s = Dragon_Curve_Sequence($n);
     
    // Printing output
    echo $s."\n";
 
// This code is contributed by mits
?>

Javascript




<script>
// Javascript code to find nth term
// of the Dragon Curve Sequence
 
// function to generate the nth term
function Dragon_Curve_Sequence(n)
{
    // first term
    let s = "1";
 
    // generating each term of the sequence
    for (let i = 2; i <= n; i++)
    {
        let temp = "1";
        let prev = '1';
        let zero = '0';
        let one = '1';
 
        // loop to generate the ith term
        for (let j = 0; j < s.length; j++)
        {
            // add character from the
            // original string
            temp = temp + s[j];
 
            // add alternate 0 and 1 in between
            if (prev == '0')
            {
                // if previous added term
                // was '0' then add '1'
                temp += one;
 
                // now current term becomes
                // previous term
                prev = one;
            }
            else
            {
                // if previous added term
                // was '1', then add '0'
                temp += zero;
 
                // now current term becomes
                // previous term
                prev = zero;
            }
        }
         
        // s becomes the ith term of the sequence
        s = temp;
    }
    return s;
}
 
// Driver code
 
    // Taking inputs
    let n = 4;
 
    // generate nth term of dragon curve sequence
    let s = Dragon_Curve_Sequence(n);
     
    // Printing output
    document.write(s + "<br>");
 
// This code is contributed by gfgking
</script>

Output: 
 

110110011100100

Complexity Analysis:

  • Time Complexity: O(n*s) where s is the length of resultant string
  • Auxiliary Space: O(s) where s is the length of resultant string 

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