Program to check if N is a Centered Triangular Number

Given an integer N, the task is to check if it is a Centered triangular number or not.

Centered triangular number is a centered polygonal number that represents a triangle with a dot in the centre and all other dots surrounding the centre in successive triangular layers . The first few Centered triangular numbers are 1, 4, 10, 19, 31, 46, 64, 85, 109, 136, …

Examples:

Input: N = 4 
Output: Yes
Input: 20 
Output: No 

Approach:

• The K^th Centered triangular number can be expressed as: 
  
  
• In order to check if the given number N can be expressed as a Centered triangular number or not, we need to check if gives an integer or not. 
  

Below is the implementation of the above approach: 

C++

// C++ implementation to check
// whether a given number is a
// Centered triangular number or not
#include <bits/stdc++.h>
using namespace std;

// Function to check if the
// number is a Centered
// Triangular Number
bool isCenteredtriangular(int N)
{
    float K = (-3
               + sqrt(24 * N - 15))
              / 6;

    // Condition for K to be
    // an integer
    return (K - (int)K) == 0;
}

// Driver Code
int main()
{
    int N = 85;
    if (isCenteredtriangular(N)) {
        cout << "Yes";
    }
    else {
        cout << "No";
    }
    return 0;
}

Java

// Java implementation to check that
// a number is a centered triangular
// number or not
import java.lang.Math;

class GFG{

// Function to check that the number
// is a centered triangular number
public static boolean isCenteredTriangular(int N)
{
    double K = (-3 + Math.sqrt(24 * N - 15)) / 6;

    // Condition to check if the number
    // is a centered triangular number
    return (K - (int)K) == 0;
}

// Driver Code
public static void main(String[] args)
{
    int N = 85;

    // Function call
    if (isCenteredTriangular(N))
    {
        System.out.println("Yes");
    }
    else
    {
        System.out.println("No");
    }
}
}

// This code is contributed by ShubhamCoder

Python3

# Python3 implementation to check
# whether a given number is a
# Centered triangular number or not
import math

# Function to check if the
# number is a Centered
# Triangular Number
def isCenteredtriangular(N):

    K = (-3 + math.sqrt(24 * N - 15)) / 6

    # Condition for K to be
    # an integer
    if (K - int(K)) == 0:
        return True

    return False

# Driver Code

# Given Number
N = 85

# Function call
if (isCenteredtriangular(N)):
    print("Yes")
else:
    print("No")

# This code is contributed by shubhamsingh10

C#

// C# implementation to check whether
// a given number is a centered
// triangular number or not
using System;

class GFG{

// Function to check if the number
// is a centered triangular number
static bool isCenteredtriangular(int N)
{
    double K = (-3 + Math.Sqrt(24 * N - 15)) / 6;

    // Condition for K to be
    // an integer
    return (K - (int)K) == 0;
}

// Driver Code
static public void Main ()
{
    int N = 85;
    if (isCenteredtriangular(N))
    {
        Console.Write("Yes");
    }
    else
    {
        Console.Write("No");
    }
}
}

// This code is contributed by shubhamsingh10

Javascript

<script>

// JavaScript implementation to check
// whether a given number is a
// Centered triangular number or not

// Function to check if the
// number is a Centered
// Triangular Number
function isCenteredtriangular(N)
{
    var K = (-3
               + Math.sqrt(24 * N - 15))
              / 6;

    // Condition for K to be
    // an integer
    return (K - parseInt(K)) == 0;
}

// Driver Code
var N = 85;
if (isCenteredtriangular(N)) {
    document.write("Yes");
}
else {
    document.write("No");
}

</script>

Output:

Yes

Time Complexity: O(logN) because inbuilt sqrt function is being used

Auxiliary Space: O(1)