Recaman’s sequence
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 is 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.
Below is a simple implementation where we store all n Recaman Sequence numbers in an array. We compute the next number using the 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 ?> |
Javascript
<script> // Javascript program to print // n-th number in Recaman's sequence // Prints first n terms of Recaman sequence function recaman(n) { // Create an array to store terms let arr = new Array(n); // First term of the sequence is always 0 arr[0] = 0; document.write(arr[0]+ " ," ); // Fill remaining terms using recursive // formula. for (let i = 1; i < n; i++) { let curr = arr[i - 1] - i; let 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; document.write(arr[i]+ ", " ); } } let n = 17; recaman(n); </script> |
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), since n extra space has been added
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 ?> |
Javascript
<script> // Javascript 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 document.write(0 + ", " ); let s = new Set(); s.add(0); // Print remaining terms using // recursive formula. let prev = 0; for (let i = 1; i< n; i++) { let curr = prev - i; // If arr[i-1] - i is negative or // already exists. if (curr < 0 || s.has(curr)) curr = prev + i; s.add(curr); document.write(curr + ", " ); prev = curr; } } // Driver code let n = 17; recaman(n); </script> |
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), since n extra space has been taken.
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