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Find a number containing N – 1 set bits at even positions from the right

  • Last Updated : 18 Aug, 2021

Given a positive integer N, the task is to find a number which contains (N – 1) set bits in its binary form at every even index (1-based) from the right.
Examples: 
 

Input: N = 2 
Output:
Binary representation of 2 is 10 which has 
1 set bit at even position from the right.
Input: N = 4 
Output: 42 
Binary representation of 42 is 101010 
 

 

Observation: If we check out the numbers in binary form then the result is something like this: 
 

nDecimal EquivalentBinary Equivalent
100
2210
3101010
442101010
517010101010

Naive Approach: As we can see in the table our binary equivalent is always adding a “10” in last of the previous string. So, we can generate a binary string which is made up of sub-string “10” concatenated N-1 times and then print its decimal equivalent.
Below is the implementation of the above approach: 
 



C++




// C++ implementation of the approach
#include <bits/stdc++.h>
using namespace std;
 
#define ll long long int
 
// Function to return the string generated
// by appending "10" n-1 times
string constructString(ll n)
{
    // Initialising string as empty
    string s = "";
    for (ll i = 0; i < n; i++) {
        s += "10";
    }
    return s;
}
 
// Function to return the decimal equivalent
// of the given binary string
ll binaryToDecimal(string n)
{
    string num = n;
    ll dec_value = 0;
 
    // Initializing base value to 1
    // i.e 2^0
    ll base = 1;
 
    ll len = num.length();
    for (ll i = len - 1; i >= 0; i--) {
        if (num[i] == '1')
            dec_value += base;
        base = base * 2;
    }
 
    return dec_value;
}
 
// Function that calls the constructString
// and binarytodecimal and returns the answer
ll findNumber(ll n)
{
    string s = constructString(n - 1);
    ll num = binaryToDecimal(s);
    return num;
}
 
// Driver code
int main()
{
    ll n = 4;
 
    cout << findNumber(n);
 
    return 0;
}

Java




// Java implementation of above approach
import java.util.*;
 
class GFG
{
 
// Function to return the String generated
// by appending "10" n-1 times
static String constructString(int n)
{
    // Initialising String as empty
    String s = "";
    for (int i = 0; i < n; i++)
    {
        s += "10";
    }
    return s;
}
 
// Function to return the decimal equivalent
// of the given binary String
static int binaryToDecimal(String n)
{
    String num = n;
    int dec_value = 0;
 
    // Initializing base value to 1
    // i.e 2^0
    int base = 1;
 
    int len = num.length();
    for (int i = len - 1; i >= 0; i--)
    {
        if (num.charAt(i) == '1')
            dec_value += base;
        base = base * 2;
    }
 
    return dec_value;
}
 
// Function that calls the constructString
// and binarytodecimal and returns the answer
static int findNumber(int n)
{
    String s = constructString(n - 1);
    int num = binaryToDecimal(s);
    return num;
}
 
// Driver code
public static void main(String[] args)
{
    int n = 4;
 
    System.out.println(findNumber(n));
}
}
 
/* This code is contributed by PrinciRaj1992 */

Python




# Python3 implementation of the approach
 
# Function to return the generated
# by appending "10" n-1 times
def constructString(n):
 
    # Initialising as empty
    s = ""
    for i in range(n):
        s += "10"
 
    return s
 
# Function to return the decimal equivaLent
# of the given binary string
def binaryToDecimal(n):
 
    num = n
    dec_value = 0
 
    # Initializing base value to 1
    # i.e 2^0
    base = 1
 
    Len = len(num)
    for i in range(Len - 1,-1,-1):
        if (num[i] == '1'):
            dec_value += base
        base = base * 2
 
 
    return dec_value
 
# Function that calls the constructString
# and binarytodecimal and returns the answer
def findNumber(n):
 
    s = constructString(n - 1)
    num = binaryToDecimal(s)
    return num
 
# Driver code
n = 4
 
print(findNumber(n))
 
# This code is contributed by mohit kumar 29

C#




// C# implementation of above approach
using System;
 
class GFG
{
     
// Function to return the String generated
// by appending "10" n-1 times
static String constructString(int n)
{
     
    // Initialising String as empty
    String s = "";
    for (int i = 0; i < n; i++)
    {
        s += "10";
    }
    return s;
}
 
// Function to return the decimal equivalent
// of the given binary String
static int binaryToDecimal(String n)
{
    String num = n;
    int dec_value = 0;
 
    // Initializing base value to 1
    // i.e 2^0
    int base_t = 1;
 
    int len = num.Length;
    for (int i = len - 1; i >= 0; i--)
    {
        if (num[i] == '1')
            dec_value = dec_value + base_t;
        base_t = base_t * 2;
    }
 
    return dec_value;
}
 
// Function that calls the constructString
// and binarytodecimal and returns the answer
static int findNumber(int n)
{
    String s = constructString(n - 1);
    int num = binaryToDecimal(s);
    return num;
}
 
// Driver code
static public void Main ()
{
    int n = 4;
    Console.Write(findNumber(n));
}
}
 
// This code is contributed by ajit

Javascript




<script>
 
// JavaScript implementation of above approach
 
// Function to return the String generated
// by appending "10" n-1 times
function constructString(n)
{
    // Initialising String as empty
    var s = "";
    for (var i = 0; i < n; i++)
    {
        s += "10";
    }
    return s;
}
 
// Function to return the decimal equivalent
// of the given binary String
function binaryToDecimal(n)
{
    var num = n;
    var dec_value = 0;
 
    // Initializing base value to 1
    // i.e 2^0
    var base = 1;
 
    var len = num.length;
    for (var i = len - 1; i >= 0; i--)
    {
        if (num.charAt(i) == '1')
            dec_value += base;
        base = base * 2;
    }
 
    return dec_value;
}
 
// Function that calls the constructString
// and binarytodecimal and returns the answer
function findNumber(n)
{
    var s = constructString(n - 1);
    var num = binaryToDecimal(s);
    return num;
}
 
// Driver code
var n = 4;
 
document.write(findNumber(n));
 
 
// This code is contributed by Amit Katiyar
 
</script>
Output: 
42

 

Efficient Approach: If we take the numbers and convert them to base 4 we can see an interesting pattern as follows: 
 

nDecimal EquivalentBinary EquivalentBase_4
1000
22102
310101022
442101010222
5170101010102222

We are actually appending “2” for every nth term in base4 i.e. for n = 7 our number in base4 would have (n – 1) i.e. 6 consecutive 2’s
Now we have to take a point in mind as we know that if we convert from any base m to base 10 i.e. decimal than the solution is (n0 * m0 + n1 * m1 + n2 * m2 + …. + n * mn). So as our base is 4 by further calculation we can found that our required number n can be found by using the deduced formula in O(1) time complexity. 
Formula: 
 

A(n) = floor((2 / 3) * (4n – 1))

Below is the implementation of the above approach: 
 

C++




// C++ implementation of the approach
#include <bits/stdc++.h>
using namespace std;
 
#define ll long long int
 
// Function to compute number
// using our deduced formula
ll findNumber(int n)
{
    // Initialize num to n-1
    ll num = n - 1;
    num = 2 * (ll)pow(4, num);
    num = floor(num / 3.0);
    return num;
}
 
// Driver code
int main()
{
    int n = 5;
    cout << findNumber(n);
 
    return 0;
}

Java




// Java implementation of the approach
import java.io.*;
 
class GFG
{
     
// Function to compute number
// using our deduced formula
static int findNumber(int n)
{
    // Initialize num to n-1
    int num = n - 1;
    num = 2 * (int)Math.pow(4, num);
    num = (int)Math.floor(num / 3.0);
    return num;
}
 
// Driver code
public static void main (String[] args)
{
    int n = 5;
    System.out.println (findNumber(n));
}
}
 
// The code is contributed by ajit.

Python3




# Python3 implementation of the approach
 
# Function to compute number
# using our deduced formula
def findNumber(n) :
     
    # Initialize num to n-1
    num = n - 1;
    num = 2 * (4 ** num);
    num = num // 3;
    return num;
 
# Driver code
if __name__ == "__main__" :
     
    n = 5;
    print(findNumber(n));
     
# This code is contributed by AnkitRai01

C#




// C# implementation of the approach
using System;
 
class GFG
{
 
// Function to compute number
// using our deduced formula
static int findNumber(int n)
{
    // Initialize num to n-1
    int num = n - 1;
    num = 2 * (int)Math.Pow(4, num);
    num = (int)Math.Floor(num / 3.0);
    return num;
}
 
// Driver code
static public void Main ()
{
         
    int n = 5;
    Console.Write(findNumber(n));
}
}
 
// The code is contributed by Tushil.

Javascript




<script>
    // Javascript implementation of the approach
     
    // Function to compute number
    // using our deduced formula
    function findNumber(n)
    {
        // Initialize num to n-1
        let num = n - 1;
        num = 2 * Math.pow(4, num);
        num = Math.floor(num / 3.0);
        return num;
    }
     
    let n = 5;
    document.write(findNumber(n));
 
</script>
Output: 
170

 

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