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Centered square numbers

In mathematics, centered square numbers are a sequence of numbers obtained by squaring the positive integers and adding consecutive odd numbers to them. These numbers are called "centered" because they can be visually represented as a square with a central dot surrounded by layers of concentric squares. The formula to calculate centered square numbers is (n^2) + ((n - 1)^2), where n represents the position in the sequence.

Let's take an example to understand this concept. Suppose we want to find the first 5 centered square numbers. We start with the number 1, which is obtained by squaring 1. The next number is calculated by squaring 2 (4) and adding the previous square, which is 0. So, the second centered square number is 4 + 0 = 4. Following the same pattern, we can calculate the third, fourth, and fifth centered square numbers as 9, 16, and 25, respectively.

Here's the algorithm to generate the first k centered square numbers:

centeredSquareNo(k):
	for n = 1 to k:
	result = (n^2) + ((n - 1)^2)
	print result
  

Code Solution

Here given code implementation process.

// C Program for
// Centered square numbers
#include <stdio.h>

void centeredSquareNo(int k)
{
	// Print all initial k centered square number
	for (int n = 1; n <= k; ++n)
	{
		// Centered square  
		// formula = (n)² + (n-1)²
      
		// CalculatedCentered square numbers
		int result = ((n *n) + ((n - 1) *(n - 1)));
      
		// Display calculated result
		printf("  %d", result);
	}
}
int main()
{
	// Centered square numbers
	// --------------------------------------
	// 1, 5, 13, 25, 41, 61, 85, 113, 145, 181, 221, 
	// 265, 313, 365, 421, 481, 545, 613, 685, 761, 
	// 841, 925, 1013, 1105, 1201, 1301, 1405, 
	// 1513, 1625, 1741, 1861, 1985, 2113, 2245, 
	// 2381, 2521, 2665, 2813, 2965, 3121, 3281, 
	// 3445, 3613, 3785, 3961, 4141 .. etc
  
	// Test 
	// k = 10
	centeredSquareNo(10);
	return 0;
}

Output

  1  5  13  25  41  61  85  113  145  181
// Java program for
// Centered square numbers
public class SquareNumber
{
	public void centeredSquareNo(int k)
	{
		// Print all initial k centered square number
		for (int n = 1; n <= k; ++n)
		{
			// Centered square  
			// formula = (n)² + (n-1)²
          
			// CalculatedCentered square numbers
			int result = ((n * n) + ((n - 1) * (n - 1)));
          
			// Display calculated result
			System.out.print(" " + result);
		}
	}
	public static void main(String[] args)
	{
		SquareNumber task = new SquareNumber();
		// Centered square numbers
		// --------------------------------------
		// 1, 5, 13, 25, 41, 61, 85, 113, 145, 181, 221, 
		// 265, 313, 365, 421, 481, 545, 613, 685, 761, 
		// 841, 925, 1013, 1105, 1201, 1301, 1405, 
		// 1513, 1625, 1741, 1861, 1985, 2113, 2245, 
		// 2381, 2521, 2665, 2813, 2965, 3121, 3281, 
		// 3445, 3613, 3785, 3961, 4141 .. etc
      
		// Test 
		// k = 10
		task.centeredSquareNo(10);
	}
}

Output

 1 5 13 25 41 61 85 113 145 181
// Include header file
#include <iostream>
using namespace std;
// C++ program for
// Centered square numbers

class SquareNumber
{
	public: void centeredSquareNo(int k)
	{
		// Print all initial k centered square number
		for (int n = 1; n <= k; ++n)
		{
			// Centered square  
			// formula = (n)² + (n-1)²
          
			// CalculatedCentered square numbers
			int result = ((n *n) + ((n - 1) *(n - 1)));
          
			// Display calculated result
			cout << " " << result;
		}
	}
};
int main()
{
	SquareNumber *task = new SquareNumber();
	// Centered square numbers
	// --------------------------------------
	// 1, 5, 13, 25, 41, 61, 85, 113, 145, 181, 221, 
	// 265, 313, 365, 421, 481, 545, 613, 685, 761, 
	// 841, 925, 1013, 1105, 1201, 1301, 1405, 
	// 1513, 1625, 1741, 1861, 1985, 2113, 2245, 
	// 2381, 2521, 2665, 2813, 2965, 3121, 3281, 
	// 3445, 3613, 3785, 3961, 4141 .. etc
  
	// Test 
	// k = 10
	task->centeredSquareNo(10);
	return 0;
}

Output

 1 5 13 25 41 61 85 113 145 181
// Include namespace system
using System;
// Csharp program for
// Centered square numbers
public class SquareNumber
{
	public void centeredSquareNo(int k)
	{
		// Print all initial k centered square number
		for (int n = 1; n <= k; ++n)
		{
			// Centered square  
			// formula = (n)² + (n-1)²
          
			// CalculatedCentered square numbers
			int result = ((n * n) + ((n - 1) * (n - 1)));
          
			// Display calculated result
			Console.Write(" " + result);
		}
	}
	public static void Main(String[] args)
	{
		SquareNumber task = new SquareNumber();
		// Centered square numbers
		// --------------------------------------
		// 1, 5, 13, 25, 41, 61, 85, 113, 145, 181, 221, 
		// 265, 313, 365, 421, 481, 545, 613, 685, 761, 
		// 841, 925, 1013, 1105, 1201, 1301, 1405, 
		// 1513, 1625, 1741, 1861, 1985, 2113, 2245, 
		// 2381, 2521, 2665, 2813, 2965, 3121, 3281, 
		// 3445, 3613, 3785, 3961, 4141 .. etc
      
		// Test 
		// k = 10
		task.centeredSquareNo(10);
	}
}

Output

 1 5 13 25 41 61 85 113 145 181
package main
import "fmt"
// Go program for
// Centered square numbers

func centeredSquareNo(k int) {
	// Print all initial k centered square number
	for n := 1 ; n <= k ; n++ {
		// Centered square  
		// formula = (n)² + (n-1)²
      
		// CalculatedCentered square numbers
		var result int = ((n * n) + ((n - 1) * (n - 1)))
      
		// Display calculated result
		fmt.Print(" ", result)
	}
}
func main() {
	
	// Centered square numbers
	// --------------------------------------
	// 1, 5, 13, 25, 41, 61, 85, 113, 145, 181, 221, 
	// 265, 313, 365, 421, 481, 545, 613, 685, 761, 
	// 841, 925, 1013, 1105, 1201, 1301, 1405, 
	// 1513, 1625, 1741, 1861, 1985, 2113, 2245, 
	// 2381, 2521, 2665, 2813, 2965, 3121, 3281, 
	// 3445, 3613, 3785, 3961, 4141 .. etc
	
	// Test 
	// k = 10
	centeredSquareNo(10)
}

Output

 1 5 13 25 41 61 85 113 145 181
<?php
// Php program for
// Centered square numbers
class SquareNumber
{
	public	function centeredSquareNo($k)
	{
		// Print all initial k centered square number
		for ($n = 1; $n <= $k; ++$n)
		{
			// Centered square  
			// formula = (n)² + (n-1)²
          
			// CalculatedCentered square numbers
			$result = (($n * $n) + (($n - 1) * ($n - 1)));
          
			// Display calculated result
			echo(" ".$result);
		}
	}
}

function main()
{
	$task = new SquareNumber();
	// Centered square numbers
	// --------------------------------------
	// 1, 5, 13, 25, 41, 61, 85, 113, 145, 181, 221, 
	// 265, 313, 365, 421, 481, 545, 613, 685, 761, 
	// 841, 925, 1013, 1105, 1201, 1301, 1405, 
	// 1513, 1625, 1741, 1861, 1985, 2113, 2245, 
	// 2381, 2521, 2665, 2813, 2965, 3121, 3281, 
	// 3445, 3613, 3785, 3961, 4141 .. etc
  
	// Test 
	// k = 10
	$task->centeredSquareNo(10);
}
main();

Output

 1 5 13 25 41 61 85 113 145 181
// Node JS program for
// Centered square numbers
class SquareNumber
{
	centeredSquareNo(k)
	{
		// Print all initial k centered square number
		for (var n = 1; n <= k; ++n)
		{
			// Centered square  
			// formula = (n)² + (n-1)²
          
			// CalculatedCentered square numbers
			var result = ((n * n) + ((n - 1) * (n - 1)));
          
			// Display calculated result
			process.stdout.write(" " + result);
		}
	}
}

function main()
{
	var task = new SquareNumber();
	// Centered square numbers
	// --------------------------------------
	// 1, 5, 13, 25, 41, 61, 85, 113, 145, 181, 221, 
	// 265, 313, 365, 421, 481, 545, 613, 685, 761, 
	// 841, 925, 1013, 1105, 1201, 1301, 1405, 
	// 1513, 1625, 1741, 1861, 1985, 2113, 2245, 
	// 2381, 2521, 2665, 2813, 2965, 3121, 3281, 
	// 3445, 3613, 3785, 3961, 4141 .. etc
  
	// Test 
	// k = 10
	task.centeredSquareNo(10);
}
main();

Output

 1 5 13 25 41 61 85 113 145 181
#  Python 3 program for
#  Centered square numbers
class SquareNumber :
	def centeredSquareNo(self, k) :
		n = 1
		#  Print all initial k centered square number
		while (n <= k) :
			#  Centered square  
			#  formula = (n)² + (n-1)²
            
			#  CalculatedCentered square numbers
			result = ((n * n) + ((n - 1) * (n - 1)))

			#  Display calculated result
			print(" ", result, end = "")
			n += 1
		
	

def main() :
	task = SquareNumber()
	#  Centered square numbers
	#  --------------------------------------
	#  1, 5, 13, 25, 41, 61, 85, 113, 145, 181, 221, 
	#  265, 313, 365, 421, 481, 545, 613, 685, 761, 
	#  841, 925, 1013, 1105, 1201, 1301, 1405, 
	#  1513, 1625, 1741, 1861, 1985, 2113, 2245, 
	#  2381, 2521, 2665, 2813, 2965, 3121, 3281, 
	#  3445, 3613, 3785, 3961, 4141 .. etc
    
	#  Test 
	#  k = 10
	task.centeredSquareNo(10)

if __name__ == "__main__": main()

Output

  1  5  13  25  41  61  85  113  145  181
#  Ruby program for
#  Centered square numbers
class SquareNumber 
	def centeredSquareNo(k) 
		n = 1
		#  Print all initial k centered square number
		while (n <= k) 
			#  Centered square  
			#  formula = (n)² + (n-1)²
            
			#  CalculatedCentered square numbers
			result = ((n * n) + ((n - 1) * (n - 1)))

			#  Display calculated result
			print(" ", result)
			n += 1
		end

	end

end

def main() 
	task = SquareNumber.new()
	#  Centered square numbers
	#  --------------------------------------
	#  1, 5, 13, 25, 41, 61, 85, 113, 145, 181, 221, 
	#  265, 313, 365, 421, 481, 545, 613, 685, 761, 
	#  841, 925, 1013, 1105, 1201, 1301, 1405, 
	#  1513, 1625, 1741, 1861, 1985, 2113, 2245, 
	#  2381, 2521, 2665, 2813, 2965, 3121, 3281, 
	#  3445, 3613, 3785, 3961, 4141 .. etc
    
	#  Test 
	#  k = 10
	task.centeredSquareNo(10)
end

main()

Output

 1 5 13 25 41 61 85 113 145 181
// Scala program for
// Centered square numbers
class SquareNumber()
{
	def centeredSquareNo(k: Int): Unit = {
		var n: Int = 1;
		// Print all initial k centered square number
		while (n <= k)
		{
			// Centered square  
			// formula = (n)² + (n-1)²
          
			// CalculatedCentered square numbers
			var result: Int = ((n * n) + ((n - 1) * (n - 1)));
          
			// Display calculated result
			print(" " + result);
			n += 1;
		}
	}
}
object Main
{
	def main(args: Array[String]): Unit = {
		var task: SquareNumber = new SquareNumber();
		// Centered square numbers
		// --------------------------------------
		// 1, 5, 13, 25, 41, 61, 85, 113, 145, 181, 221, 
		// 265, 313, 365, 421, 481, 545, 613, 685, 761, 
		// 841, 925, 1013, 1105, 1201, 1301, 1405, 
		// 1513, 1625, 1741, 1861, 1985, 2113, 2245, 
		// 2381, 2521, 2665, 2813, 2965, 3121, 3281, 
		// 3445, 3613, 3785, 3961, 4141 .. etc
		// Test 
		// k = 10
		task.centeredSquareNo(10);
	}
}

Output

 1 5 13 25 41 61 85 113 145 181
// Swift 4 program for
// Centered square numbers
class SquareNumber
{
	func centeredSquareNo(_ k: Int)
	{
		var n: Int = 1;
		// Print all initial k centered square number
		while (n <= k)
		{
			// Centered square  
			// formula = (n)² + (n-1)²
          
			// CalculatedCentered square numbers
			let result: Int = ((n * n) + ((n - 1) * (n - 1)));
          
			// Display calculated result
			print(" ", result, terminator: "");
			n += 1;
		}
	}
}
func main()
{
	let task: SquareNumber = SquareNumber();
	// Centered square numbers
	// --------------------------------------
	// 1, 5, 13, 25, 41, 61, 85, 113, 145, 181, 221, 
	// 265, 313, 365, 421, 481, 545, 613, 685, 761, 
	// 841, 925, 1013, 1105, 1201, 1301, 1405, 
	// 1513, 1625, 1741, 1861, 1985, 2113, 2245, 
	// 2381, 2521, 2665, 2813, 2965, 3121, 3281, 
	// 3445, 3613, 3785, 3961, 4141 .. etc
  
	// Test 
	// k = 10
	task.centeredSquareNo(10);
}
main();

Output

  1  5  13  25  41  61  85  113  145  181
// Kotlin program for
// Centered square numbers
class SquareNumber
{
	fun centeredSquareNo(k: Int): Unit
	{
		var n: Int = 1;
		// Print all initial k centered square number
		while (n <= k)
		{
			// Centered square  
			// formula = (n)² + (n-1)²
          
			// CalculatedCentered square numbers
			val result: Int = ((n * n) + ((n - 1) * (n - 1)));
          
			// Display calculated result
			print(" " + result);
			n += 1;
		}
	}
}
fun main(args: Array < String > ): Unit
{
	val task: SquareNumber = SquareNumber();
	// Centered square numbers
	// --------------------------------------
	// 1, 5, 13, 25, 41, 61, 85, 113, 145, 181, 221, 
	// 265, 313, 365, 421, 481, 545, 613, 685, 761, 
	// 841, 925, 1013, 1105, 1201, 1301, 1405, 
	// 1513, 1625, 1741, 1861, 1985, 2113, 2245, 
	// 2381, 2521, 2665, 2813, 2965, 3121, 3281, 
	// 3445, 3613, 3785, 3961, 4141 .. etc
  
	// Test 
	// k = 10
	task.centeredSquareNo(10);
}

Output

 1 5 13 25 41 61 85 113 145 181

Time complexity

The time complexity of the algorithm is O(k), as it performs a constant amount of work for each value from 1 to k. The space complexity is constant as well, as it only requires a single variable to store the calculated result.





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