Program to check if N is a Centered Octagonal Number

Given an integer N, the task is to check if it is a Centered Octagonal number or not. If the number N is an Centered Octagonal Number then print “Yes” else print “No”.

Centered Octagonal number represents an octagon with a dot in the centre and others dots surrounding the centre dot in the successive octagonal layer.The first few Centered Octagonal numbers are 1, 9, 25, 49, 81, 121, 169, 225, 289, 361 …

Examples:

Input: N = 9
Output: Yes
Explanation:
Second Centered Octagonal number is 9.

Input: 16
Output: No



Approach:

  1. The Kth term of the Centered Octagonal number is given as

    K^{th} Term =  4*K^{2} - 4*K + 1

  2. As we have to check that the given number can be expressed as a Centered Octagonal Number or not. This can be checked as follows –

    => N =  {4*K^{2} - 4*K + 1}
    => K = \frac{1 + \sqrt{N}}{2}

  3. If the value of K calculated using the above formula is an integer, then N is a Centered Octagonal Number.
  4. Else N is not a Centered Octagonal Number.

Below is the implementation of the above approach:

C++

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// C++ program for the above approach
#include <bits/stdc++.h>
  
using namespace std;
  
// Function to check if the number N
// is a Centered Octagonal number
bool isCenteredOctagonal(int N)
{
    float n
        = (1 + sqrt(N))
          / 2;
  
    // Condition to check if the number
    // is a Centered Octagonal number
    return (n - (int)n) == 0;
}
  
// Driver Code
int main()
{
    // Given Number
    int N = 9;
  
    // Function call
    if (isCenteredOctagonal(N)) {
        cout << "Yes";
    }
    else {
        cout << "No";
    }
    return 0;
}

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Java

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// Java program for the above approach
import java.util.*;
  
class GFG{
  
// Function to check if the number N
// is a centered octagonal number
static boolean isCenteredOctagonal(int N)
{
    float n = (float) ((1 + Math.sqrt(N)) / 2);
  
    // Condition to check if the number
    // is a centered octagonal number
    return (n - (int)n) == 0;
}
  
// Driver Code
public static void main(String[] args)
{
      
    // Given Number
    int N = 9;
  
    // Function call
    if (isCenteredOctagonal(N))
    {
        System.out.print("Yes");
    }
    else
    {
        System.out.print("No");
    }
}
}
  
// This code is contributed by sapnasingh4991

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Python3

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# Python3 program for the above approach
import numpy as np
  
# Function to check if the number N
# is a centered octagonal number
def isCenteredOctagonal(N):
  
    n = (1 + np.sqrt(N)) / 2
  
    # Condition to check if N 
    # is a centered octagonal number
    return (n - int(n)) == 0
  
# Driver Code 
N = 9
  
# Function call 
if (isCenteredOctagonal(N)):
    print("Yes"
else:
    print("No")
  
# This code is contributed by PratikBasu

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C#

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// C# program for the above approach
using System;
  
class GFG{
  
// Function to check if the number N
// is a centered octagonal number
static bool isCenteredOctagonal(int N)
{
    float n = (float) ((1 + Math.Sqrt(N)) / 2);
  
    // Condition to check if the number
    // is a centered octagonal number
    return (n - (int)n) == 0;
}
  
// Driver Code
public static void Main(string[] args)
{
      
    // Given Number
    int N = 9;
  
    // Function call
    if (isCenteredOctagonal(N))
    {
        Console.Write("Yes");
    }
    else
    {
        Console.Write("No");
    }
}
}
  
// This code is contributed by rutvik_56

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Output:

Yes

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