Centered Square Number

Given a number n , the task is to find n^th Centered Square Number.

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. N^th Centered square number can be calculated by using formula n² + (n-1)².

Examples :

Input : n = 2
Output : 5

Input : n = 9
Output : 145

1. 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 :

   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.

   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))

   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.

   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. ?>

   Output :

   7th Centered square number: 85
   

2. 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 number
     N = Given Number
     
     Solve for n:
     H(n) = N
     n * n + ((n - 1) * (n - 1)) = N
     
     Applying Shridharacharya Formula
     The 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:

   CPP

   
   

   
   
   

   #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

   Python3

   
   

   
   
   

   # 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