程序检查N是否是居中的三边形数字

给定一个数字N ，任务是检查N是否是一个居中的三边形数字。如果数字N是居中的三边形数字，则打印“是”，否则打印“否” 。

Centered tridecagonal number represents a dot at the center and other dots surrounding the center dot in the successive tridecagonal(13 sided polygon) layer. The first few Centered tridecagonal numbers are 1, 14, 40, 79 … 

例子：

Input: N = 14 
Output: Yes 
Explanation: 
Second Centered tridecagonal number is 14.

Input: N = 30 
Output: No   

方法：

1.中心三边形数的第K^个项为
$K^{th} Term = \frac{{13*K^{2} - 13*K + 2}}{2}$

2.由于我们必须检查给定的数字是否可以表示为居中的三边形数字。可以检查如下：

=> $N = \frac{{13*K^{2} - 13*K + 2}}{2}$
=> $K = \frac{13 + \sqrt{104*N + 65}}{26}$

3.如果使用上述公式计算出的K的值为整数，则N为居中的三边形数。

4.否则，数字N不是居中的三边形数字。

下面是上述方法的实现：

C++

// C++ program for the above approach
#include 
using namespace std;
 
// Function to check if the number N
// is a Centered tridecagonal number
bool isCenteredtridecagonal(int N)
{
    float n
        = (13 + sqrt(104 * N + 65))
          / 26;
 
    // Condition to check if the N
    // is a Centered tridecagonal number
    return (n - (int)n) == 0;
}
 
// Driver Code
int main()
{
    // Given Number
    int N = 14;
 
    // Function call
    if (isCenteredtridecagonal(N)) {
        cout << "Yes";
    }
    else {
        cout << "No";
    }
    return 0;
}

Java

// Java program for the above approach
class GFG{
 
// Function to check if the number N
// is a centered tridecagonal number
static boolean isCenteredtridecagonal(int N)
{
    float n = (float) ((13 + Math.sqrt(104 * N +
                                       65)) / 26);
 
    // Condition to check if the N
    // is a centered tridecagonal number
    return (n - (int)n) == 0;
}
 
// Driver Code
public static void main(String[] args)
{
     
    // Given Number
    int N = 14;
 
    // Function call
    if (isCenteredtridecagonal(N))
    {
        System.out.print("Yes");
    }
    else
    {
        System.out.print("No");
    }
}
}
 
// This code is contributed by sapnasingh4991

Python3

# Python3 program for the above approach
import numpy as np
 
# Function to check if the number N
# is a centered tridecagonal number
def isCenteredtridecagonal(N):
 
    n = (13 + np.sqrt(104 * N + 65)) / 26
 
    # Condition to check if N
    # is centered tridecagonal number
    return (n - int(n)) == 0
 
# Driver Code
N = 14
 
# Function call
if (isCenteredtridecagonal(N)):
    print ("Yes")
else:
    print ("No")
 
# This code is contributed by PratikBasu

C#

// C# program for the above approach
using System;
 
class GFG{
 
// Function to check if the number N
// is a centered tridecagonal number
static bool isCenteredtridecagonal(int N)
{
    float n = (float) ((13 + Math.Sqrt(104 * N +
                                       65)) / 26);
 
    // Condition to check if the N
    // is a centered tridecagonal number
    return (n - (int)n) == 0;
}
 
// Driver Code
public static void Main(string[] args)
{
     
    // Given Number
    int N = 14;
 
    // Function call
    if (isCenteredtridecagonal(N))
    {
        Console.Write("Yes");
    }
    else
    {
        Console.Write("No");
    }
}
}
 
// This code is contributed by rutvik_56

Javascript

// Javascript program for the above approach
 
// Function to check if the number N
// is a Centered tridecagonal number
function isCenteredtridecagonal(N)
{
    let n
        = (13 + Math.sqrt(104 * N + 65))
          / 26;
 
    // Condition to check if the N
    // is a Centered tridecagonal number
    return (n - parseInt(n)) == 0;
}
 
// Driver Code
// Given Number
let N = 14;
 
// Function call
if (isCenteredtridecagonal(N)) {
    document.write("Yes");
}
else {
    document.write("No");
}
 
// This code is contributed by subham348.

输出：

Yes

时间复杂度： O(1)

辅助空间： O(1)