Open In App

Find nth term of the Dragon Curve Sequence

Improve
Improve
Like Article
Like
Save
Share
Report

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 


Last Updated : 19 Sep, 2022
Like Article
Save Article
Previous
Next
Share your thoughts in the comments
Similar Reads