Recaman's sequence - GeeksforGeeks

Recaman’s sequence

Difficulty Level : Easy
Last Updated : 21 Aug, 2019

Given an integer n. Print first n elements of Recaman’s sequence.

Examples:

Input : n = 6
Output : 0, 1, 3, 6, 2, 7

Input  : n = 17
Output : 0, 1, 3, 6, 2, 7, 13, 20, 12, 21, 
         11, 22, 10, 23, 9, 24, 8

It is basically a function with domain and co-domain as natural numbers and 0. It is recursively defined as below:
Specifically, let a(n) denote the (n+1)-th term. (0 being already there).
The rule says:

a(0) = 0,
if n > 0 and the number is not 
   already included in the sequence,
     a(n) = a(n - 1) - n 
else 
     a(n) = a(n-1) + n.

Recommended: Please solve it on “PRACTICE” first, before moving on to the solution.

Below is simple implementation where we store all n Recaman Sequence numbers in an array. We compute next number using recursive formula mentioned above.

C++

// C++ program to print n-th number in Recaman's  
// sequence 
#include <bits/stdc++.h> 
using namespace std; 
  
// Prints first n terms of Recaman sequence 
int recaman(int n) 
{ 
    // Create an array to store terms 
    int arr[n]; 
  
    // First term of the sequence is always 0 
    arr[0] = 0; 
    printf("%d, ", arr[0]); 
  
    // Fill remaining terms using recursive 
    // formula. 
    for (int i=1; i< n; i++) 
    { 
        int curr = arr[i-1] - i; 
        int j; 
        for (j = 0; j < i; j++) 
        { 
            // If arr[i-1] - i is negative or 
            // already exists. 
            if ((arr[j] == curr) || curr < 0) 
            { 
                curr = arr[i-1] + i; 
                break; 
            } 
        } 
  
        arr[i] = curr; 
        printf("%d, ", arr[i]); 
    } 
} 
  
// Driver code 
int main() 
{ 
    int n = 17; 
    recaman(n); 
    return 0; 
} 

Java

// Java program to print n-th number in Recaman's  
// sequence 
import java.io.*; 
  
class GFG { 
      
    // Prints first n terms of Recaman sequence 
    static void recaman(int n) 
    { 
        // Create an array to store terms 
        int arr[] = new int[n]; 
      
        // First term of the sequence is always 0 
        arr[0] = 0; 
        System.out.print(arr[0]+" ,"); 
      
        // Fill remaining terms using recursive 
        // formula. 
        for (int i = 1; i < n; i++) 
        { 
            int curr = arr[i - 1] - i; 
            int j; 
            for (j = 0; j < i; j++) 
            { 
                // If arr[i-1] - i is negative or 
                // already exists. 
                if ((arr[j] == curr) || curr < 0) 
                { 
                    curr = arr[i - 1] + i; 
                    break; 
                } 
            } 
      
            arr[i] = curr; 
            System.out.print(arr[i]+", "); 
              
        } 
    } 
      
    // Driver code 
    public static void main (String[] args)  
    { 
        int n = 17; 
        recaman(n); 
  
    } 
} 
  
// This code is contributed by vt_m 

Python 3

# Python 3 program to print n-th 
# number in Recaman's sequence 
  
# Prints first n terms of Recaman 
# sequence 
def recaman(n): 
  
    # Create an array to store terms 
    arr = [0] * n 
  
    # First term of the sequence 
    # is always 0 
    arr[0] = 0
    print(arr[0], end=", ") 
  
    # Fill remaining terms using 
    # recursive formula. 
    for i in range(1, n): 
      
        curr = arr[i-1] - i 
        for j in range(0, i): 
          
            # If arr[i-1] - i is 
            # negative or already 
            # exists. 
            if ((arr[j] == curr) or curr < 0): 
                curr = arr[i-1] + i 
                break
              
        arr[i] = curr 
        print(arr[i], end=", ") 
  
# Driver code 
n = 17
  
recaman(n) 
  
# This code is contributed by Smitha. 

C#

// C# program to print n-th number in Recaman's  
// sequence 
using System; 
  
class GFG { 
      
    // Prints first n terms of Recaman sequence 
    static void recaman(int n) 
    { 
        // Create an array to store terms 
        int []arr = new int[n]; 
      
        // First term of the sequence is always 0 
        arr[0] = 0; 
        Console.Write(arr[0]+" ,"); 
      
        // Fill remaining terms using recursive 
        // formula. 
        for (int i = 1; i < n; i++) 
        { 
            int curr = arr[i - 1] - i; 
            int j; 
            for (j = 0; j < i; j++) 
            { 
                // If arr[i-1] - i is negative or 
                // already exists. 
                if ((arr[j] == curr) || curr < 0) 
                { 
                    curr = arr[i - 1] + i; 
                    break; 
                } 
            } 
      
            arr[i] = curr; 
        Console.Write(arr[i]+", "); 
              
        } 
    } 
      
    // Driver code 
    public static void Main ()  
    { 
        int n = 17; 
        recaman(n); 
  
    } 
} 
  
// This code is contributed by vt_m. 

PHP

<?php 
// PHP program to print n-th 
// number in Recaman's sequence 
  
// Prints first n terms  
// of Recaman sequence 
function recaman($n) 
{ 
      
    // First term of the  
    // sequence is always 0 
    $arr[0] = 0; 
    echo $arr[0], ", "; 
  
    // Fill remaining terms  
    // using recursive formula. 
    for ($i = 1; $i < $n; $i++) 
    { 
            $curr = $arr[$i - 1] - $i; 
            $j; 
        for ($j = 0; $j < $i; $j++) 
        { 
              
            // If arr[i-1] - i 
            // is negative or 
            // already exists. 
            if (($arr[$j] == $curr) || $curr < 0) 
            { 
                $curr = $arr[$i-1] + $i; 
                break; 
            } 
        } 
  
        $arr[$i] = $curr; 
        echo $arr[$i], ", "; 
    } 
} 
  
    // Driver Code 
    $n = 17; 
    recaman($n); 
      
// This code is contributed by Ajit 
?> 

Output:

0, 1, 3, 6, 2, 7, 13, 20, 12, 21, 11, 22, 10, 23, 9, 24, 8,

Time Complexity : O(n²)
Auxiliary Space : O(n)

Optimizations :
We can use hashing to store previously computed values and can make this program work in O(n) time.

C++

// C++ program to print n-th number in Recaman's  
// sequence 
#include <bits/stdc++.h> 
using namespace std; 
  
// Prints first n terms of Recaman sequence 
void recaman(int n) 
{ 
    if (n <= 0) 
      return; 
  
    // Print first term and store it in a hash  
    printf("%d, ", 0); 
    unordered_set<int> s; 
    s.insert(0); 
  
    // Print remaining terms using recursive 
    // formula. 
    int prev = 0; 
    for (int i=1; i< n; i++) 
    { 
        int curr = prev - i; 
  
        // If arr[i-1] - i is negative or 
        // already exists. 
        if (curr < 0 || s.find(curr) != s.end()) 
           curr = prev + i; 
  
        s.insert(curr); 
  
        printf("%d, ", curr); 
        prev = curr; 
    } 
} 
  
// Driver code 
int main() 
{ 
    int n = 17; 
    recaman(n); 
    return 0; 
} 

Java

// Java program to print n-th number  
// in Recaman's sequence 
import java.util.*; 
  
class GFG 
{ 
  
// Prints first n terms of Recaman sequence 
static void recaman(int n) 
{ 
    if (n <= 0) 
    return; 
  
    // Print first term and store it in a hash  
    System.out.printf("%d, ", 0); 
    HashSet<Integer> s = new HashSet<Integer>(); 
    s.add(0); 
  
    // Print remaining terms using  
    // recursive formula. 
    int prev = 0; 
    for (int i = 1; i< n; i++) 
    { 
        int curr = prev - i; 
  
        // If arr[i-1] - i is negative or 
        // already exists. 
        if (curr < 0 || s.contains(curr)) 
            curr = prev + i; 
  
        s.add(curr); 
  
        System.out.printf("%d, ", curr); 
        prev = curr; 
    } 
} 
  
// Driver code 
public static void main(String[] args) 
{ 
    int n = 17; 
    recaman(n); 
} 
} 
  
// This code is contributed by Rajput-Ji 

Python3

# Python3 program to print n-th number in 
# Recaman's sequence 
  
# Prints first n terms of Recaman sequence 
def recaman(n): 
  
    if(n <= 0): 
        return
  
    # Print first term and store it in a hash 
    print(0, ",", end='') 
    s = set([]) 
    s.add(0) 
  
    # Print remaining terms using recursive 
    # formula. 
    prev = 0
    for i in range(1, n): 
  
        curr = prev - i 
  
        # If arr[i-1] - i is negative or 
        # already exists. 
        if(curr < 0 or curr in s): 
            curr = prev + i 
  
        s.add(curr) 
  
        print(curr, ",", end='') 
        prev = curr 
  
# Driver code 
if __name__=='__main__': 
    n = 17
    recaman(n) 
  
# This code is contributed by 
# Sanjit_Prasad 

C#

// C# program to print n-th number  
// in Recaman's sequence 
using System; 
using System.Collections.Generic; 
  
class GFG 
{ 
  
// Prints first n terms of Recaman sequence 
static void recaman(int n) 
{ 
    if (n <= 0) 
    return; 
  
    // Print first term and store it in a hash  
    Console.Write("{0}, ", 0); 
    HashSet<int> s = new HashSet<int>(); 
    s.Add(0); 
  
    // Print remaining terms using  
    // recursive formula. 
    int prev = 0; 
    for (int i = 1; i < n; i++) 
    { 
        int curr = prev - i; 
  
        // If arr[i-1] - i is negative or 
        // already exists. 
        if (curr < 0 || s.Contains(curr)) 
            curr = prev + i; 
  
        s.Add(curr); 
  
        Console.Write("{0}, ", curr); 
        prev = curr; 
    } 
} 
  
// Driver code 
public static void Main(String[] args) 
{ 
    int n = 17; 
    recaman(n); 
} 
} 
  
// This code is contributed by Princi Singh 

PHP

<?php 
// PHP program to print n-th number in  
// Recaman's sequence  
  
// Prints first n terms of Recaman sequence  
function recaman($n)  
{ 
    if($n <= 0)  
        return; 
  
    // Print first term and store  
    // it in a hash  
    print("0, ");  
    $s = array();  
    array_push($s, 0);  
  
    // Print remaining terms using recursive  
    // formula.  
    $prev = 0; 
    for ($i = 1; $i < $n; $i++) 
    {  
        $curr = $prev - $i;  
  
        // If arr[i-1] - i is negative or  
        // already exists.  
        if($curr < 0 or in_array($curr, $s))  
            $curr = $prev + $i;  
  
        array_push($s, $curr);  
  
        print($curr.", ");  
        $prev = $curr; 
    } 
          
} 
  
// Driver code  
$n = 17; 
recaman($n);  
  
// This code is contributed by chandan_jnu 
?> 

Output:

0, 1, 3, 6, 2, 7, 13, 20, 12, 21, 11, 22, 10, 23, 9, 24, 8,

Time Complexity : O(n)
Auxiliary Space : O(n)

This article is contributed by Kishlay Verma. If you like GeeksforGeeks and would like to contribute, you can also write an article using contribute.geeksforgeeks.org or mail your article to contribute@geeksforgeeks.org. See your article appearing on the GeeksforGeeks main page and help other Geeks.

Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above.

Attention reader! Don’t stop learning now. Get hold of all the important DSA concepts with the DSA Self Paced Course at a student-friendly price and become industry ready.

My Personal Notes

Related Articles

Recommended: Please solve it on “PRACTICE” first, before moving on to the solution.

C++

Java

Python 3

C#

PHP

C++

Java

Python3

C#

PHP