# Newman-Conway Sequence

Newman-Conway Sequence is the one which generates the following integer sequence.
1 1 2 2 3 4 4 4 5 6 7 7…

In mathematical terms, the sequence P(n) of Newman-Conway numbers is defined by recurrence relation

```P(n) = P(P(n - 1)) + P(n - P(n - 1))
```

with seed values P(1) = 1 and P(2) = 1

Given a number n, print n-th number in Newman-Conway Sequence.

Examples :

```Input : n = 2
Output : 1

Input : n = 10
Output : 6
```

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

Method 1 (Use Recursion) :
A simple approach is direct recursive implementation of above recurrence relation.

## C++

 `// C++ program for n-th  ` `// element of Newman-Conway Sequence ` `#include ` `using` `namespace` `std; ` ` `  `// Recursive Function to find the n-th element ` `int` `sequence(``int` `n) ` `{ ` `    ``if` `(n == 1 || n == 2) ` `        ``return` `1; ` `    ``else` `        ``return` `sequence(sequence(n - 1))  ` `                ``+ sequence(n - sequence(n - 1)); ` `} ` ` `  `// Driver Program ` `int` `main() ` `{ ` `    ``int` `n = 10; ` `    ``cout << sequence(n); ` `    ``return` `0; ` `} `

## Java

 `// Java program to find nth ` `// element of Newman-Conway Sequence ` `import` `java.io.*; ` ` `  `class` `GFG { ` `     `  `    ``// Recursion to find  ` `    ``// n-th element ` `    ``static` `int` `sequence(``int` `n) ` `    ``{ ` `        ``if` `(n == ``1` `|| n == ``2``) ` `            ``return` `1``; ` `        ``else` `            ``return` `sequence(sequence(n - ``1``))  ` `                  ``+ sequence(n - sequence(n - ``1``)); ` `    ``} ` `      `  `    ``// Driver Program ` `    ``public` `static` `void` `main(String args[]) ` `    ``{ ` `        ``int` `n = ``10``; ` `        ``System.out.println(sequence(n)); ` `    ``} ` `} ` ` `  `/*This code is contributed by Nikita Tiwari.*/`

## Python

 `# Recursive function to find the n-th  ` `# element of sequence ` `def` `sequence(n): ` `    ``if` `n ``=``=` `1` `or` `n ``=``=` `2``: ` `        ``return` `1` `    ``else``: ` `        ``return` `sequence(sequence(n``-``1``)) ``+` `sequence(n``-``sequence(n``-``1``)); ` `         `  `# Driver code ` `def` `main(): ` `    ``n ``=` `10` `    ``print` `sequence(n) ` `     `  `if` `__name__ ``=``=` `'__main__'``: ` `    ``main() `

## C#

 `// C# program to find nth element ` `// of Newman-Conway Sequence ` `using` `System; ` ` `  `class` `GFG { ` `     `  `    ``// Recursion to find  ` `    ``// n-th element ` `    ``static` `int` `sequence(``int` `n) ` `    ``{ ` `        ``if` `(n == 1 || n == 2) ` `            ``return` `1; ` `        ``else` `            ``return` `sequence(sequence(n - 1)) + sequence ` `                           ``(n - sequence(n - 1)); ` `    ``} ` `     `  `    ``// Driver code ` `    ``public` `static` `void` `Main() ` `    ``{ ` `        ``int` `n = 10; ` `        ``Console.Write(sequence(n)); ` `    ``} ` `} ` ` `  `// This code is contributed by Nitin Mittal. `

## PHP

 ` `

Output :

`6`

Method 2 (Use Dynamic Programming) :
We can avoid repeated work done in method 1 by storing the values in the sequence in an array.

## C++

 `// C++ program to find the n-th element of  ` `// Newman-Conway Sequence ` `#include ` `using` `namespace` `std; ` ` `  `// Function to find the n-th element ` `int` `sequence(``int` `n) ` `{ ` `    ``// Declare array to store sequence ` `    ``int` `f[n + 1]; ` `    ``int` `i; ` `    ``f = 0; ` `    ``f = 1; ` `    ``f = 1; ` ` `  `    ``for` `(i = 3; i <= n; i++)  ` `        ``f[i] = f[f[i - 1]] + f[i - f[i - 1]];     ` ` `  `    ``return` `f[n]; ` `} ` ` `  `// Driver Program ` `int` `main() ` `{ ` `    ``int` `n = 10; ` `    ``cout << sequence(n); ` `    ``return` `0; ` `} `

## Java

 `// JAVA Code for Newman-Conway Sequence ` `import` `java.util.*; ` ` `  `class` `GFG { ` `     `  `    ``// Function to find the n-th element ` `    ``static` `int` `sequence(``int` `n) ` `    ``{ ` `        ``// Declare array to store sequence ` `        ``int` `f[] = ``new` `int``[n + ``1``]; ` `        ``f[``0``] = ``0``; ` `        ``f[``1``] = ``1``; ` `        ``f[``2``] = ``1``; ` ` `  `        ``int` `i; ` `      `  `        ``for` `(i = ``3``; i <= n; i++)  ` `            ``f[i] = f[f[i - ``1``]] + ` `                        ``f[i - f[i - ``1``]];     ` `      `  `        ``return` `f[n]; ` `    ``} ` `     `  `    ``/* Driver program to test above function */` `    ``public` `static` `void` `main(String[] args)  ` `    ``{ ` `         ``int` `n = ``10``; ` `         ``System.out.println(sequence(n)); ` ` `  `    ``} ` `} ` ` `  `// This code is contributed by Arnav Kr. Mandal. `

## Python

 `''' Python program to find the n-th element of  ` `    ``Newman-Conway Sequence'''` ` `  `# To declare array import module array ` `import` `array ` `def` `sequence(n): ` `    ``f ``=` `array.array(``'i'``, [``0``, ``1``, ``1``]) ` ` `  `    ``# To store values of sequence in array ` `    ``for` `i ``in` `range``(``3``, n ``+` `1``): ` `        ``r ``=` `f[f[i``-``1``]]``+``f[i``-``f[i``-``1``]] ` `        ``f.append(r); ` ` `  `    ``return` `r ` ` `  `# Driver code ` `def` `main(): ` `    ``n ``=` `10` `    ``print` `sequence(n) ` `     `  `if` `__name__ ``=``=` `'__main__'``: ` `    ``main() `

## C#

 `// C# Code for Newman-Conway Sequence ` `using` `System; ` ` `  `class` `GFG { ` `     `  `    ``// Function to find the n-th element ` `    ``static` `int` `sequence(``int` `n) ` `    ``{ ` `        ``// Declare array to store sequence ` `        ``int` `[]f = ``new` `int``[n + 1]; ` `        ``f = 0; ` `        ``f = 1; ` `        ``f = 1; ` ` `  `        ``int` `i; ` `     `  `        ``for` `(i = 3; i <= n; i++)  ` `            ``f[i] = f[f[i - 1]] + ` `                   ``f[i - f[i - 1]];  ` `     `  `        ``return` `f[n]; ` `    ``} ` `     `  `    ``// Driver Code ` `    ``public` `static` `void` `Main()  ` `    ``{ ` `        ``int` `n = 10; ` `        ``Console.Write(sequence(n)); ` ` `  `    ``} ` `} ` ` `  `// This code is contributed by Nitin Mittal. `

## PHP

 ` `

Output :

`6`

Time Complexity : O(n)

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