Recaman's sequence - GeeksforGeeks

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)

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My Personal Notes

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

C++

Java

Python 3

C#

PHP

C++

Java

Python3

C#

PHP

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