Newman-Conway Sequence is the one that 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 the 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
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 elementint sequence(int n){ if (n == 1 || n == 2) return 1; else return sequence(sequence(n - 1)) + sequence(n - sequence(n - 1));}// Driver Programint main(){ int n = 10; cout << sequence(n); return 0;} |
Java
// Java program to find nth// element of Newman-Conway Sequenceimport 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 sequencedef sequence(n): if n == 1 or n == 2: return 1 else: return sequence(sequence(n-1)) + sequence(n-sequence(n-1)); # Driver codedef main(): n = 10 print sequence(n) if __name__ == '__main__': main() |
C#
// C# program to find nth element// of Newman-Conway Sequenceusing 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 elementfunction 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.?> |
Javascript
<script>// JavaScript program to find nth// element of Newman-Conway Sequence// Recursion to find // n-th elementfunction sequence(n){ if (n == 1 || n == 2) return 1; else return sequence(sequence(n - 1)) + sequence(n - sequence(n - 1));}// Driver codelet n = 10;document.write(sequence(n));// This code is contributed by souravghosh0416 </script> |
Output :
6
Time complexity: O(n)
Auxiliary Space: O(n)
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 elementint 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 Programint main(){ int n = 10; cout << sequence(n); return 0;} |
Java
// JAVA Code for Newman-Conway Sequenceimport 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 arrayimport arraydef 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 codedef main(): n = 10 print sequence(n) if __name__ == '__main__': main() |
C#
// C# Code for Newman-Conway Sequenceusing 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 elementfunction 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.?> |
Javascript
<script>// Javascript program to find the n-th element// of Newman-Conway Sequence// Function to find// the n-th elementfunction sequence(n){ // Declare array to // store sequence let i; let f = []; f[0] = 0; f[1] = 1; f[2] = 1; for (let i = 3; i <= n; i++) f[i] = f[f[i - 1]] + f[i - f[i - 1]]; return f[n];}// Driver Codelet n = 10;document.write(sequence(n));// This code is contributed by gfgking.</script> |
Output :
6
Time complexity: O(n)
Auxiliary Space: O(n)
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