Newman-Conway Sequence

Difficulty Level : Easy
Last Updated : 25 Apr, 2018

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 <bits/stdc++.h> 
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

<?php 
// PHP program for n-th element  
// of Newman-Conway Sequence 
  
// Recursive Function to  
// find the n-th element 
function sequence($n) 
{ 
    if ($n == 1 || $n == 2) 
        return 1; 
    else
        return sequence(sequence($n - 1))+  
               sequence($n - sequence($n - 1)); 
} 
  
// Driver Code 
$n = 10; 
echo(sequence($n)); 
  
// This code is contributed by Ajit. 
?> 

Output :

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 <bits/stdc++.h> 
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] = 0; 
    f[1] = 1; 
    f[2] = 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] = 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 Code 
    public static void Main()  
    { 
        int n = 10; 
        Console.Write(sequence(n)); 
  
    } 
} 
  
// This code is contributed by Nitin Mittal. 

PHP

<?php 
// PHP program to find the n-th element   
// of Newman-Conway Sequence 
  
// Function to find  
// the n-th element 
function sequence($n) 
{ 
      
    // Declare array to  
    // store sequence 
    $i; 
    $f[0] = 0; 
    $f[1] = 1; 
    $f[2] = 1; 
  
    for ($i = 3; $i <= $n; $i++)  
        $f[$i] = $f[$f[$i - 1]] +  
                 $f[$i - $f[$i - 1]];  
  
    return $f[$n]; 
} 
  
// Driver Code 
$n = 10; 
echo(sequence($n)); 
  
// This code is contributed by Ajit. 
?> 

Output :

Time Complexity : O(n)

References : https://archive.lib.msu.edu/crcmath/math/math/n/n078.htm

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