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Centered Square Number

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Given a number n, the task is to find nth Centered Square Number. A centered Square Number is a centered figurate number that gives the number of dots in a square with a dot in the center and all other dots surrounding the center dot in successive square layers. Nth Centered square number can be calculated by using the formula n2 + (n-1)2.

Centered-Square-Number

Examples :

Input: n = 2
Output: 5

Input: n = 9
Output: 145

Finding n-th Centered Square Number:

If we take a closer look, we can notice that the n-th Centered Square Number can be seen as the sum of two consecutive square numbers (1 dot, 4 dots, 9 dots, 16 dots, etc). We can find n-th Centered Square Number using below formula.

n-th Centered Square Number = n2 + (n-1)2

Below is the implementation of the above approach:

C++
// C++ program to find nth
// Centered square number.
#include <bits/stdc++.h>

using namespace std;

// Function to calculate Centered
// square number function
int centered_square_num(int n)
{
    // Formula to calculate nth
    // Centered square number
    return n * n + ((n - 1) * (n - 1));
}

// Driver Code
int main()
{
    int n = 7;
    cout << n << "th Centered square number: ";
    cout << centered_square_num(n);
    return 0;
}
Java
// Java program to find nth Centered square
// number
import java.io.*;

class GFG {

    // Function to calculate Centered
    // square number function
    static int centered_square_num(int n)
    {
        // Formula to calculate nth
        // Centered square number
        return n * n + ((n - 1) * (n - 1));
    }
    
    // Driver Code
    public static void main (String[] args) 
    {
        int n = 7;
        System.out.print( n + "th Centered"
                       + " square number: "
                 + centered_square_num(n));
    }
}

// This code is contributed by anuj_67.
C#
// C# program to find nth
// Centered square number.
using System;

public class GFG {

    // Function to calculate Centered
    // square number function
    static int centered_square_num(int n)
    {
        // Formula to calculate nth
        // Centered square number
        return n * n + ((n - 1) * (n - 1));
    }
    
    // Driver Code

    static public void Main (){
    int n = 7;
    Console.WriteLine( n + "th Centered"
                    + " square number: "
               + centered_square_num(n));
    }
}

// This code is contributed by anuj_67.
JavaScript
// Function to calculate centered square number
function centeredSquareNum(n) {
    // Formula to calculate nth centered square number
    return n * n + ((n - 1) * (n - 1));
}

// Main function
function main() {
    const n = 7;
    console.log(`${n}th Centered square number: ${centeredSquareNum(n)}`);
}

// Calling the main function
main();
PHP
<?php
// PHP program to find nth
// Centered square number

// Function to calculate Centered
// square number function
function centered_square_num( $n)
{
    // Formula to calculate nth
    // Centered square number
    return $n * $n + (($n - 1) * 
                      ($n - 1));
}

// Driver Code
$n = 7;
echo $n , "th Centered square number: ";
echo centered_square_num($n);

// This code is contributed by anuj_67.
?>
Python3
# Python program to find nth
# Centered square number.


# Function to calculate Centered
# square number function
def centered_square_num(n):

    # Formula to calculate nth
    # Centered square number
    return n * n + ((n - 1) * (n - 1))


# Driver Code
n = 7
print("%sth Centered square number: " %n,
                  centered_square_num(n))
    

Output
7th Centered square number: 85

Check if N is centred-square-number or not:

The first few centered-square-number numbers are:

1,5,13,25,41,61,85,113,145,181,…………

Since the nth centered-square-number number is given by

H(n) = n * n + ((n - 1) * (n - 1))

The formula indicates that the n-th centred-square-number number depends quadratically on n. Therefore, try to find the positive integral root of N = H(n) equation.

H(n) = nth centered-square-number numberN = Given NumberSolve for n:H(n) = Nn * n + ((n - 1) * (n - 1)) = NApplying Shridharacharya FormulaThe positive root of equation (i)n = (9 + sqrt(36*N+45))/18; 

After obtaining n, check if it is an integer or not. n is an integer if n – floor(n) is 0

Below is the implementation of the above approach:

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

bool centeredSquare_number(int N) 
{     
    float n = (9 + sqrt(36*N+45))/18; 
    return (n - (int) n) == 0; 
} 

int main() 
{ 
    int i = 13; 
    cout<<centeredSquare_number(i); 
    return 0; 
} 
Java
// Java Code implementation of the above approach 
class GFG { 
    
    static int centeredSquare_number(int N) 
    {     
        float n = (9 + (float)Math.sqrt(36*N+45))/18; 
        if (n - (int) n == 0) 
            return 1; 
        else
            return 0; 
    } 
    
    // Driver code 
    public static void main (String[] args) 
    { 
        int i = 13; 
        System.out.println(centeredSquare_number(i)); 
    } 
    
} 

// This code is contributed by Yash_R 
Python
# Python3 implementation of the above approach 
from math import sqrt 

def centeredSquare_number(N) : 

    n = (9 + sqrt(36 * N + 45))/18; 
    if (n - int(n)) == 0 : 
        return 1
    else : 
        return 0

# Driver Code 
if __name__ == "__main__" : 

    i = 13; 
    print(centeredSquare_number(i)); 

# This code is contributed by Yash_R 
C#
// C# Code implementation of the above approach 
using System; 

class GFG { 
    
    static int centeredSquare_number(int N) 
    {     
        float n = (9 + (float)Math.Sqrt(36 * N + 45))/18; 
        if (n - (int) n == 0) 
            return 1; 
        else
            return 0; 
    } 
    
    // Driver code 
    public static void Main (String[] args) 
    { 
        int i = 13; 
        Console.WriteLine(centeredSquare_number(i)); 
    } 
    
} 

// This code is contributed by Yash_R 

Output
0

Reference:.https://en.wikipedia.org/wiki/Centered_square_number



Last Updated : 28 Mar, 2024
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