Program to find the nth Kynea number

Given a positive integer n, the task is to find nth Kynea number. 
Kynea number: In mathematics, a Kynea number is a positive integer of the form: 

where n is a positive integer.
The equivalent formula for nth Kynea number is: 

The first few Kynea number are: 

7, 23, 79, 287, 1087, 4223, 16639, 66047, 263167, 1050623, 4198399, …..

Examples:

Input: 2
Output: 23
             Putting n = 2 in formula,
            = 4² + 2 ²⁺¹ – 1
            = 16 + 8 -1
            = 23  

Method 1: A Simple Solution is to find out the nth number by putting the value of n in the formula 

Below is the implementation of the above approach:

C++

// CPP code to find nth Kynea number

#include <bits/stdc++.h>
using namespace std;

// Function to calculate nth kynea number
long nthKyneaNumber(int n)
{

    // Calculate nth kynea number
    // using formula ((2^n + 1)^2 ) -2

    // Firstly calculate 2^n + 1
    n = (1 << n) + 1;

    // Now calculate (2^n + 1)^2
    n = n * n;

    // Now calculate ((2^n + 1)^2 ) - 2
    n = n - 2;

    // return nth Kynea number
    return n;
}

// Driver Program
int main()
{
    int n = 8;

    // print nth kynea number
    cout << nthKyneaNumber(n);

    return 0;
}

Java

// JAVA code to find nth Kynea number

class GFG {

    // Function to calculate nth kynea number
    static long nthKyneaNumber(int n)
    {

        // Calculate nth kynea number
        // using formula ((2^n + 1)^2 ) -2

        // Firstly calculate 2^n + 1
        n = (1 << n) + 1;

        // Now calculate (2^n + 1)^2
        n = n * n;

        // Now calculate ((2^n + 1)^2 ) - 2
        n = n - 2;

        // return nth Kynea number
        return n;
    }

    // Driver Program
    public static void main(String[] args)
    {
        int n = 2;

        // print nth kynea number
        System.out.println(nthKyneaNumber(n));
    }
}

Python

# Python code to find nth Kynea number

# Function to calculate nth kynea number
def nthKyneaNumber( n):

    # Calculate nth kynea number
    # using formula ((2 ^ n + 1)^2 ) -2

    # Firstly calculate 2 ^ n + 1
    n = ( 1 << n) + 1

    # Now calculate (2 ^ n + 1)^2
    n = n * n

    # Now calculate ((2 ^ n + 1)^2 ) - 2
    n = n-2


    # return nth Kynea number
    return n



# Driver Code
n = 2

# print nth kynea number
print(nthKyneaNumber(n))

C#

// C# code to find nth Kynea number

using System;
class GFG {

    // Function to calculate nth kynea number
    static long nthKyneaNumber(int n)
    {

        // Calculate nth kynea number
        // using formula ((2^n + 1)^2 ) -2

        // Firstly calculate 2^n + 1
        n = (1 << n) + 1;

        // Now calculate (2^n + 1)^2
        n = n * n;

        // Now calculate ((2^n + 1)^2 ) - 2
        n = n - 2;

        // return nth Kynea number
        return n;
    }

    // Driver Program
    public static void Main()
    {
        int n = 2;

        // print nth kynea number
        Console.WriteLine(nthKyneaNumber(n));
    }
}

PHP

<?php
// PHP code to find nth Kynea number

// Function to calculate nth kynea number
function nthKyneaNumber($n)
{

    // Calculate nth kynea number
    // using formula ((2^n + 1)^2 ) -2

    // Firstly calculate 2^n + 1
    $n = (1 << $n) + 1;

    // Now calculate (2^n + 1)^2
    $n = $n * $n;

    // Now calculate ((2^n + 1)^2 ) - 2
    $n = $n - 2;

    // return nth Kynea number
    return $n;
}

// Driver Code
$n = 8;

// print nth kynea number
echo nthKyneaNumber($n);

// This code is contributed
// by Akanksha Rai(Abby_akku)

Javascript

<script>


// Javascript code to find nth Kynea number

// Function to calculate nth kynea number
function nthKyneaNumber(n)
{

    // Calculate nth kynea number
    // using formula ((2^n + 1)^2 ) -2

    // Firstly calculate 2^n + 1
    n = (1 << n) + 1;

    // Now calculate (2^n + 1)^2
    n = n * n;

    // Now calculate ((2^n + 1)^2 ) - 2
    n = n - 2;

    // return nth Kynea number
    return n;
}

// Driver Code
let n = 8;

// print nth kynea number
document.write( nthKyneaNumber(n));

// This code is contributed by bobby

</script>

Output:

66047

Time Complexity: O(1), only constant operations are being used.
Auxiliary Space: O(1), as no extra space is required.

Method 2: This solution is based on the fact that every Kynea number follows a specific pattern in their binary representation. nth Kynea number can be represented in binary as a single leading one followed by exactly n-1 consecutive 0’s, followed by n+1 consecutive 1’s.

Example:

23 is 2nd kynea number
It can be represented in binary as 10111 
(Single leading one, followed by n - 1 ( i.e 2-1=1 ) consecutive 0's, 
followed by n + 1 ( i.e 2 + 1 = 3 ) consecutive 1's.)

n	nth Kynea Number	Binary Representation
1	7	111
2	23	10111
3	79	1001111
4	287	100011111
5	1087	10000111111
6	4223	1000001111111

Observing the binary pattern of Kynea number in above table, the nth Kynea number can be easily calculated using the formula: 

Example:

Input: n = 3
Output: 79
              Using formula,
              = 2⁶ + 2⁴ -1
              = 64 + 15 
              = 79      

Below is the implementation of the above approach

C++

// CPP code to find nth Kynea number

#include <bits/stdc++.h>
using namespace std;

// Function to calculate nth kynea number
long nthKyneaNumber(int n)
{

    // Calculate nth kynea number
    return ((1 << (2 * n)) + (1 << (n + 1)) - 1);
}

// Driver Program
int main()
{
    int n = 2;

    // print nth kynea number
    cout << nthKyneaNumber(n);

    return 0;
}

Java

// JAVA code to find nth Kynea number

class GFG {

    // Function to calculate nth kynea number
    static long nthKyneaNumber(int n)
    {

        // Calculate nth kynea number
        return ((1 << (2 * n)) + (1 << (n + 1)) - 1);
    }

    // Driver Program
    public static void main(String[] args)
    {
        int n = 2;

        // print nth kynea number
        System.out.println(nthKyneaNumber(n));
    }
}

Python

# Python code to find nth Kynea number

# Function to calculate nth kynea number
def nthKyneaNumber( n):

    # Calculate nth kynea number
    return (( 1 << (2 * n)) + ( 1 << (n + 1)) -1 )


# Driver Code
n = 2

# print nth kynea number
print(nthKyneaNumber(n))

C#

// C# code to find nth Kynea number

using System;
class GFG {

    // Function to calculate nth kynea number
    static long nthKyneaNumber(int n)
    {

        // Calculate nth kynea number
        return ((1 << (2 * n)) + (1 << (n + 1)) - 1);
    }

    // Driver Program
    public static void Main()
    {
        int n = 2;

        // print nth kynea number
        Console.WriteLine(nthKyneaNumber(n));
    }
}

PHP

<?php
// PHP code to find nth Kynea number

// Function to calculate
// nth kynea number
function nthKyneaNumber($n)
{

    // Calculate nth kynea number
    return ((1 << (2 * $n)) +
            (1 << ($n + 1)) - 1);
}

// Driver Code
$n = 2;

// print nth kynea number
echo nthKyneaNumber($n);

// This code is contributed
// by anuj_67
?>

Javascript

<script>

// Javascript code to find nth Kynea number

// Function to calculate
// nth kynea number
function nthKyneaNumber(n)
{
    // Calculate nth kynea number
    return ((1 << (2 * n)) +
            (1 << (n + 1)) - 1);
}

// Driver Code
let n = 2;

// print nth kynea number
document.write( nthKyneaNumber(n));

// This code is contributed by bobby

</script>

Output:

23

Time Complexity: O(1), only constant operations are being used.
Auxiliary Space: O(1), as no extra space is required.