Newman-Conway Sequence
Last Updated :
14 Feb, 2023
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++
#include <bits/stdc++.h>
using namespace std;
int sequence(int n)
{
if (n == 1 || n == 2)
return 1;
else
return sequence(sequence(n - 1))
+ sequence(n - sequence(n - 1));
}
int main()
{
int n = 10;
cout << sequence(n);
return 0;
}
|
Java
import java.io.*;
class GFG {
static int sequence(int n)
{
if (n == 1 || n == 2)
return 1;
else
return sequence(sequence(n - 1))
+ sequence(n - sequence(n - 1));
}
public static void main(String args[])
{
int n = 10;
System.out.println(sequence(n));
}
}
|
Python
def sequence(n):
if n == 1 or n == 2:
return 1
else:
return sequence(sequence(n-1)) + sequence(n-sequence(n-1));
def main():
n = 10
print sequence(n)
if __name__ == '__main__':
main()
|
C#
using System;
class GFG {
static int sequence(int n)
{
if (n == 1 || n == 2)
return 1;
else
return sequence(sequence(n - 1)) + sequence
(n - sequence(n - 1));
}
public static void Main()
{
int n = 10;
Console.Write(sequence(n));
}
}
|
PHP
<?php
function sequence($n)
{
if ($n == 1 || $n == 2)
return 1;
else
return sequence(sequence($n - 1))+
sequence($n - sequence($n - 1));
}
$n = 10;
echo(sequence($n));
?>
|
Javascript
<script>
function sequence(n)
{
if (n == 1 || n == 2)
return 1;
else
return sequence(sequence(n - 1)) +
sequence(n - sequence(n - 1));
}
let n = 10;
document.write(sequence(n));
</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++
#include <bits/stdc++.h>
using namespace std;
int sequence(int n)
{
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];
}
int main()
{
int n = 10;
cout << sequence(n);
return 0;
}
|
Java
import java.util.*;
class GFG {
static int sequence(int n)
{
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];
}
public static void main(String[] args)
{
int n = 10;
System.out.println(sequence(n));
}
}
|
Python
import array
def sequence(n):
f = array.array('i', [0, 1, 1])
for i in range(3, n + 1):
r = f[f[i-1]]+f[i-f[i-1]]
f.append(r);
return r
def main():
n = 10
print sequence(n)
if __name__ == '__main__':
main()
|
C#
using System;
class GFG {
static int sequence(int n)
{
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];
}
public static void Main()
{
int n = 10;
Console.Write(sequence(n));
}
}
|
PHP
<?php
function sequence($n)
{
$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];
}
$n = 10;
echo(sequence($n));
?>
|
Javascript
<script>
function sequence(n)
{
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];
}
let n = 10;
document.write(sequence(n));
</script>
|
Output :
6
Time complexity: O(n)
Auxiliary Space: O(n)
Like Article
Suggest improvement
Share your thoughts in the comments
Please Login to comment...