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Moser-de Bruijn Sequence

  • Difficulty Level : Easy
  • Last Updated : 26 May, 2021

Given an integer ‘n’, print the first ‘n’ terms of the Moser-de Bruijn Sequence.
The Moser-de Bruijn sequence is the sequence obtained by adding up the distinct powers of the number 4 (For example, 1, 4, 16, 64, etc). 

Examples : 

Input : 5
Output : 0 1 4 5 16

Input : 10
Output : 0 1 4 5 16 17 20 21 64 65

It is observed that the terms of the sequence follow the recurrence relation :  

1) S(2 * n) = 4 * S(n)
2) S(2 * n + 1) = 4 * S(n) + 1
with S(0) = 0 and S(1) = 1

It may be noted here that any number which is the sum of non-distinct powers of 4 is not a part of the sequence. For example, 8 is not a part of the sequence because it is formed as the sum of non-distinct powers of 4, which are 4 and 4. 
Thus, any number which is not a power of 4 and is present in the sequence must be the sum of the distinct powers of 4. 
For example, 21 is a part of the sequence, even though it is not a power of 4, because it is the sum of the distinct powers of 4, which are 1, 4, and 16.

Employ the recurrence relation discussed above to generate the sequence efficiently. 



C++




// CPP code to generate first 'n' terms
// of the Moser-de Bruijn Sequence
#include <bits/stdc++.h>
using namespace std;
 
// Function to generate nth term
// of Moser-de Bruijn Sequence
int gen(int n)
{
    // S(0) = 0
    if (n == 0)
        return 0;
     
    // S(1) = 1
    else if (n == 1)
        return 1;
     
    // S(2 * n) = 4 * S(n)
    else if (n % 2 == 0)
        return 4 * gen(n / 2);
     
    // S(2 * n + 1) = 4 * S(n) + 1
    else if (n % 2 == 1)
        return 4 * gen(n / 2) + 1;
}
 
// Generating the first 'n' terms
// of Moser-de Bruijn Sequence
void moserDeBruijn(int n)
{
    for (int i = 0; i < n; i++)
        cout << gen(i) << " ";
    cout << "\n";
}
 
// Driver Code
int main()
{
    int n = 15;
    cout << "First " << n << " terms of "
         << "Moser-de Bruijn Sequence : \n";
    moserDeBruijn(n);
    return 0;
}

Java




// Java code to generate first 'n' terms
// of the Moser-de Bruijn Sequence
 
class GFG
{
     
// Function to generate nth term
// of Moser-de Bruijn Sequence
public static int gen(int n)
{
     
    // S(0) = 0
    if (n == 0)
        return 0;
     
    // S(1) = 1
    else if (n == 1)
        return 1;
     
    // S(2 * n) = 4 * S(n)
    else if (n % 2 == 0)
        return 4 * gen(n / 2);
     
    // S(2 * n + 1) = 4 * S(n) + 1
    else if (n % 2 == 1)
        return 4 * gen(n / 2) + 1;
    return 0;
}
 
// Generating the first 'n' terms
// of Moser-de Bruijn Sequence
public static void moserDeBruijn(int n)
{
    for (int i = 0; i < n; i++)
        System.out.print(gen(i) + " ");
    System.out.println();
}
 
// Driver Code
public static void main(String args[])
{
    int n = 15;
    System.out.println("First " + n +
                       " terms of " +
      "Moser-de Bruijn Sequence : ");
    moserDeBruijn(n);
}
}
 
// This code is contributed by JaideepPyne.

Python3




# Python code to generate first
# 'n' terms of the Moser-de
# Bruijn Sequence
 
# Function to generate nth term
# of Moser-de Bruijn Sequence
def gen(n):
 
    # S(0) = 0
    if n == 0:
        return 0
 
    # S(1) = 1
    elif n ==1:
        return 1
 
    # S(2 * n) = 4 * S(n)
    elif n % 2 ==0:
        return 4 * gen(n // 2)
 
    # S(2 * n + 1) = 4 * S(n) + 1
    elif n % 2 == 1:
        return 4 * gen(n // 2) +1
 
# Generating the first 'n' terms
# of Moser-de Bruijn Sequence
def moserDeBruijn(n):
    for i in range(n):
        print(gen(i), end = " ")
 
# Driver Program
n = 15
print("First", n, "terms of ",
       "Moser-de Brujn Sequence:")
moserDeBruijn(n)
 
# This code is contributed by Shrikant13

C#




// C# code to generate first 'n' terms
// of the Moser-de Bruijn Sequence
using System;
 
class GFG {
     
// Function to generate nth term
// of Moser-de Bruijn Sequence
public static int gen(int n)
{
     
    // S(0) = 0
    if (n == 0)
        return 0;
     
    // S(1) = 1
    else if (n == 1)
        return 1;
     
    // S(2 * n) = 4 * S(n)
    else if (n % 2 == 0)
        return 4 * gen(n / 2);
     
    // S(2 * n + 1) = 4 * S(n) + 1
    else if (n % 2 == 1)
        return 4 * gen(n / 2) + 1;
    return 0;
}
 
// Generating the first 'n' terms
// of Moser-de Bruijn Sequence
public static void moserDeBruijn(int n)
{
    for (int i = 0; i < n; i++)
        Console.Write(gen(i) + " ");
        Console.WriteLine();
}
 
// Driver Code
public static void Main()
{
    int n = 15;
    Console.WriteLine("First " + n +
                    " terms of " +
    "Moser-de Bruijn Sequence : ");
    moserDeBruijn(n);
}
}
 
// This code is contributed by anuj_67.

PHP




<?php
// PHP code to generate first 'n' terms
// of the Moser-de Bruijn Sequence
 
// Function to generate nth term
// of Moser-de Bruijn Sequence
function gen($n)
{
     
    // S(0) = 0
    if ($n == 0)
        return 0;
     
    // S(1) = 1
    else if ($n == 1)
        return 1;
     
    // S(2 * n) = 4 * S(n)
    else if ($n % 2 == 0)
        return 4 * gen($n / 2);
     
    // S(2 * n + 1) = 4 * S(n) + 1
    else if ($n % 2 == 1)
        return 4 * gen($n / 2) + 1;
}
 
// Generating the first 'n' terms
// of Moser-de Bruijn Sequence
function moserDeBruijn($n)
{
    for ($i = 0; $i < $n; $i++)
        echo(gen($i) . " ");
    echo("\n");
}
 
// Driver Code
$n = 15;
echo("First " . $n . " terms of " .
  "Moser-de Bruijn Sequence : \n");
echo(moserDeBruijn($n));
 
// This code is contributed by Ajit.
?>

Javascript




<script>
 
// Javascript code to generate first 'n' terms
// of the Moser-de Bruijn Sequence
 
// Function to generate nth term
// of Moser-de Bruijn Sequence
function gen(n)
{
 
    // S(0) = 0
    if (n == 0)
        return 0;
 
    // S(1) = 1
    else if (n == 1)
        return 1;
 
    // S(2 * n) = 4 * S(n)
    else if (n % 2 == 0)
        return 4 * gen(parseInt(n / 2, 10));
 
    // S(2 * n + 1) = 4 * S(n) + 1
    else if (n % 2 == 1)
        return 4 * gen(parseInt(n / 2, 10)) + 1;
         
    return 0;
}
 
// Generating the first 'n' terms
// of Moser-de Bruijn Sequence
function moserDeBruijn(n)
{
    for(let i = 0; i < n; i++)
        document.write(gen(i) + " ");
         
    document.write("</br>");
}
 
// Driver code
let n = 15;
document.write("First " + n + " terms of " +
               "Moser-de Bruijn Sequence : " + "</br>");
moserDeBruijn(n);
 
// This code is contributed by decode2207
 
</script>

Output : 

First 15 terms of Moser-de Bruijn Sequence : 
0 1 4 5 16 17 20 21 64 65 68 69 80 81 84

Dynamic Programming Implementation: 

C++




// CPP code to generate first 'n' terms
// of the Moser-de Bruijn Sequence
#include <bits/stdc++.h>
using namespace std;
 
// Function to generate nth term
// of Moser-de Bruijn Sequence
int gen(int n)
{
    int S[n+1];
 
    S[0] = 0;
    S[1] = 1;
 
    for (int i = 2; i <= n; i++)
    {   
        // S(2 * n) = 4 * S(n)
        if (i % 2 == 0)
           S[i] = 4 * S[i / 2];
     
        // S(2 * n + 1) = 4 * S(n) + 1
        else
           S[i] = 4 * S[i / 2] + 1;
    }
     
    return S[n];
}
 
// Generating the first 'n' terms
// of Moser-de Bruijn Sequence
void moserDeBruijn(int n)
{
    for (int i = 0; i < n; i++)
        cout << gen(i) << " ";
    cout << "\n";
}
 
// Driver Code
int main()
{
    int n = 15;
    cout << "First " << n << " terms of "
         << "Moser-de Bruijn Sequence : \n";
    moserDeBruijn(n);
    return 0;
}

Java




// Java code to generate first 'n' terms
// of the Moser-de Bruijn Sequence
 
class GFG
{
     
// Function to generate nth term
// of Moser-de Bruijn Sequence
static int gen(int n)
{
    int []S = new int [n + 1];
 
    S[0] = 0;
    if(n != 0)
        S[1] = 1;
 
    for (int i = 2; i <= n; i++)
    {
         
        // S(2 * n) = 4 * S(n)
        if (i % 2 == 0)
        S[i] = 4 * S[i / 2];
     
        // S(2 * n + 1) = 4 * S(n) + 1
        else
        S[i] = 4 * S[i/2] + 1;
    }
     
    return S[n];
}
 
// Generating the first 'n' terms
// of Moser-de Bruijn Sequence
static void moserDeBruijn(int n)
{
    for (int i = 0; i < n; i++)
        System.out.print(gen(i)+" ");
}
 
// Driver Code
public static void main(String[] args)
{
    int n = 15;
    System.out.println("First " + n +
                       " terms of " +
      "Moser-de Bruijn Sequence : ");
    moserDeBruijn(n);
}
}
 
// This code is contributed by
// Smitha Dinesh Semwal.

python3




# python3 code to generate first 'n' terms
# of the Moser-de Bruijn Sequence
 
# Function to generate nth term
# of Moser-de Bruijn Sequence
def gen( n ):
    S = [0, 1]
    for i in range(2, n+1):
         
        # S(2 * n) = 4 * S(n)
        if i % 2 == 0:
            S.append(4 * S[int(i / 2)]);
             
        # S(2 * n + 1) = 4 * S(n) + 1
        else:
            S.append(4 * S[int(i / 2)] + 1);
    z = S[n];
    return z;
 
# Generating the first 'n' terms
# of Moser-de Bruijn Sequence
def moserDeBruijn(n):
    for i in range(n):
        print(gen(i), end = " ")
 
# Driver Code
n = 15
print("First", n, "terms of ",
    "Moser-de Brujn Sequence:")
moserDeBruijn(n)
 
# This code is contributed by mits.

C#




// C# code to generate first 'n' terms
// of the Moser-de Bruijn Sequence
using System;
 
class GFG
{
     
// Function to generate nth term
// of Moser-de Bruijn Sequence
static int gen(int n)
{
    int []S = new int [n + 1];
 
    S[0] = 0;
    if(n != 0)
        S[1] = 1;
 
    for (int i = 2; i <= n; i++)
    {
         
        // S(2 * n) = 4 * S(n)
        if (i % 2 == 0)
        S[i] = 4 * S[i / 2];
     
        // S(2 * n + 1) = 4 * S(n) + 1
        else
        S[i] = 4 * S[i/2] + 1;
    }
     
    return S[n];
}
 
// Generating the first 'n' terms
// of Moser-de Bruijn Sequence
static void moserDeBruijn(int n)
{
    for (int i = 0; i < n; i++)
        Console.Write(gen(i)+" ");
}
 
// Driver Code
public static void Main()
{
    int n = 15;
    Console.WriteLine("First " + n +
                      " terms of " +
     "Moser-de Bruijn Sequence : ");
    moserDeBruijn(n);
}
}
 
// This code is contributed by
// Smitha Dinesh Semwal.

PHP




<?php
// PHP code to generate first 'n' terms
// of the Moser-de Bruijn Sequence
 
// Function to generate nth term
// of Moser-de Bruijn Sequence
function gen( $n)
{
    $S = array();
 
    $S[0] = 0;
    $S[1] = 1;
 
    for ( $i = 2; $i <= $n; $i++)
    {
        // S(2 * n) = 4 * S(n)
        if ($i % 2 == 0)
        $S[$i] = 4 * $S[$i / 2];
     
        // S(2 * n + 1) = 4 * S(n) + 1
        else
        $S[$i] = 4 * $S[$i/2] + 1;
    }
     
    return $S[$n];
}
 
// Generating the first 'n' terms
// of Moser-de Bruijn Sequence
function moserDeBruijn( $n)
{
    for ( $i = 0; $i < $n; $i++)
        echo gen($i) , " ";
    echo "\n";
}
 
// Driver Code
$n = 15;
echo "First " ,$n ," terms of "
    , "Moser-de Bruijn Sequence : \n";
moserDeBruijn($n);
 
// This code is contributed by anuj_67.
?>

Javascript




<script>
    // Javascript code to generate first 'n' terms
    // of the Moser-de Bruijn Sequence
     
    // Function to generate nth term
    // of Moser-de Bruijn Sequence
    function gen(n)
    {
        let S = new Array(n + 1);
        S.fill(0);
 
        S[0] = 0;
        if(n != 0)
            S[1] = 1;
 
        for (let i = 2; i <= n; i++)
        {
 
            // S(2 * n) = 4 * S(n)
            if (i % 2 == 0)
                S[i] = 4 * S[parseInt(i / 2, 10)];
 
            // S(2 * n + 1) = 4 * S(n) + 1
            else
                S[i] = 4 * S[parseInt(i / 2, 10)] + 1;
        }
 
        return S[n];
    }
 
    // Generating the first 'n' terms
    // of Moser-de Bruijn Sequence
    function moserDeBruijn(n)
    {
        for (let i = 0; i < n; i++)
            document.write(gen(i)+" ");
    }
     
    let n = 15;
    document.write("First " + n +
                      " terms of " +
     "Moser-de Bruijn Sequence : " + "</br>");
    moserDeBruijn(n);
 
</script>

Output : 

First 15 terms of Moser-de Bruijn Sequence : 
0 1 4 5 16 17 20 21 64 65 68 69 80 81 84

 

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