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 
.
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 termstring 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 programint 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 Sequenceimport java.util.*;class solution{// function to generate the nth termstatic 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 programpublic 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 termdef 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 Coden = 4# generate nth term of # dragon curve sequences = Dragon_Curve_Sequence(n)# Printing outputprint(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 Codepublic 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 termfunction 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 termfunction 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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