# 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++ program to print n-th number in Recaman's ` `// sequence` `#include ` `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;` `    ``printf``(``"%d, "``, arr);`   `    ``// 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 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 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# 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;` `        ``Console.Write(arr+``" ,"``);` `    `  `        ``// 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.`

 ``

 ``

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++ program to print n-th number in Recaman's ` `// sequence` `#include ` `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 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 s = ``new` `HashSet();` `    ``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 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# 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`

 ``

 ``

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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