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Sum of first N Star Numbers

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Given a number N, the task is to find the sum of the first N star numbers
The first few star numbers are 1, 13, 37, 73,..
Examples: 

Input: N = 2 
Output: 14 
Explanation: 1, 13 are the first two star numbers.

Input: N = 3 
Output: 51 

Approach 1: 

  1. Nth Star number is given as 6*n^2 - 6*n + 1
  2. Run a loop from 1 to N, to find the first N star numbers.
  3. Add all the above-calculated star numbers.
  4. Return the sum.

Below is the implementation of the above approach:

C++

// C++ program to find the sum of
// the first N Star Number
#include <bits/stdc++.h>
using namespace std;
 
// Function to find the N-th
// Star Number
int star_num(int n)
{
     
    // Formula to calculate nth
    // Star Number
    return (6 * n * n - 6 * n + 1);
}
 
// Function to find the sum of the
// first N Star Number
int sum_star_num(int n)
{
     
    // Variable to store the sum
    int summ = 0;
     
    // Iterating from 1 to N
    for(int i = 1; i < n + 1; i++)
    {
         
        // Finding the sum
        summ += star_num(i);
    }
    return summ;
}
 
// Driver code
int main()
{
    int n = 3;
     
    cout << sum_star_num(n);
}
 
// This code is contributed by spp____

                    

Java

// Java program to find the sum of
// the first N Star Number
class GFG{
 
// Function to find the N-th
// Star Number
static int star_num(int n)
{
     
    // Formula to calculate nth
    // Star Number
    return (6 * n * n - 6 * n + 1);
}
 
// Function to find the sum of the
// first N Star Number
static int sum_star_num(int n)
{
     
    // Variable to store the sum
    int summ = 0;
     
    // Iterating from 1 to N
    for(int i = 1; i < n + 1; i++)
    {
         
        // Finding the sum
        summ += star_num(i);
    }
    return summ;
}
 
// Driver code
public static void main(String[] args)
{
    int n = 3;
     
    System.out.println(sum_star_num(n));
}
}
 
// This code is contributed by rock_cool

                    

Python3

# Python3 program to find the
# sum of the first N 
# star numbers
 
# Function to find the
# N-th star
# number
def star_num(n):
  
    # Formula to calculate 
    # nth star
    # number
    return (6 * n * n - 6 * n + 1)
    
# Function to find the sum of
# the first N star numbers
def sum_star_num(n) :
     
    # Variable to store
    # the sum
    summ = 0
     
    # Iterating in the range
    # 1 to N
    for i in range(1, n + 1):
        summ += star_num(i)
     
    return summ
   
# Driver code
n = 3
print(sum_star_num(n))

                    

C#

// C# program to find the sum of
// the first N Star Number
using System;
class GFG{
 
// Function to find the N-th
// Star Number
static int star_num(int n)
{
     
    // Formula to calculate nth
    // Star Number
    return (6 * n * n - 6 * n + 1);
}
 
// Function to find the sum of the
// first N Star Number
static int sum_star_num(int n)
{
     
    // Variable to store the sum
    int summ = 0;
     
    // Iterating from 1 to N
    for(int i = 1; i < n + 1; i++)
    {
         
        // Finding the sum
        summ += star_num(i);
    }
    return summ;
}
 
// Driver code
public static void Main(String[] args)
{
    int n = 3;
     
    Console.WriteLine(sum_star_num(n));
}
}
 
// This code is contributed by gauravrajput1

                    

Javascript

<script>
 
    // Javascript program to find the sum of    
    // the first N Star Number
     
    // Function to find the N-th
    // Star Number
    function star_num(n)
    {
 
        // Formula to calculate nth
        // Star Number
        return (6 * n * n - 6 * n + 1);
    }
 
    // Function to find the sum of the
    // first N Star Number
    function sum_star_num(n)
    {
 
        // Variable to store the sum
        let summ = 0;
 
        // Iterating from 1 to N
        for(let i = 1; i < n + 1; i++)
        {
 
            // Finding the sum
            summ += star_num(i);
        }
        return summ;
    }
     
    let n = 3;
       
    document.write(sum_star_num(n));
     
</script>

                    

Output
51

Time complexity: O(N).
Auxiliary Space: O(1)

Efficient Approach: 
 

  • We already know \sum n = \frac{n*(n+1)}{2}       \sum n^2 = \frac{n*(n+1)*(2n+1)}{6}       \sum n^3 = \frac{n*(n+1)}{2}^2       and \sum 1 = n
  • Nth star number is given as 6*n^2 - 6*n + 1
  • So, the sum of the first N Star Numbers is \sum 6*n^2 - 6*n + 1
    Sum = 6*\sum n^2 - 6*\sum n + \sum 1
    Sum = 6*\frac{n*(n+1)*(2n+1)}{6} - 6*\frac{n*(n+1)}{2} + n
    Sum = 2*n*(n+1)*(n-1) + n
  • Calculate the sum and return.


Below is the implementation of the above approach: 

C++

// C++ program to find the
// sum of the first N
// star numbers
#include <bits/stdc++.h>
 
using namespace std;
 
// Function to find the
// sum of the first N
// star number
int sum_star_num(int n)
{
     
    // Variable to store
    // the sum
    int summ = 2 * n * (n + 1) * (n - 1) + n;
     
    return summ;
}
 
// Driver code
int main()
{
    int n = 3;
     
    cout << sum_star_num(n);
    return 0;
}
 
// This code is contributed by Amit Katiyar

                    

Java

// Java program to find the
// sum of the first N 
// star numbers
class GFG{
     
    // Function to find the
    // sum of the first N
    // star number
    static int sum_star_num(int n)
    {
 
        // Variable to store
        // the sum
        int summ = 2 * n * (n + 1) * (n - 1) + n;
 
        return summ;
    }
 
    // Driver code
    public static void main(String[] args)
    {
        int n = 3;
        System.out.println(sum_star_num(n));
    }
}
 
// This code is contributed by PrinciRaj1992

                    

Python3

# Python3 program to find the
# sum of the first N 
# star numbers
 
# Function to find the
# sum of the first N
# star number
def sum_star_num(n) :
     
    # Variable to store
    # the sum
    summ = 2 * n*(n + 1)*(n-1) + n
     
    return summ
   
# Driver code
n = 3
print(sum_star_num(n))

                    

C#

// C# program to find the
// sum of the first N
// star numbers
using System;
 
class GFG{
     
// Function to find the
// sum of the first N
// star number
static int sum_star_num(int n)
{
 
    // Variable to store
    // the sum
    int summ = 2 * n * (n + 1) * (n - 1) + n;
 
    return summ;
}
 
// Driver code
public static void Main(String[] args)
{
    int n = 3;
     
    Console.WriteLine(sum_star_num(n));
}
}
 
// This code is contributed by PrinciRaj1992

                    

Javascript

<script>
// Javascript program to find the
// sum of the first N
// star numbers
 
// Function to find the
// sum of the first N
// star number
function sum_star_num(n)
{
     
    // Variable to store
    // the sum
    let summ = 2 * n * (n + 1) * (n - 1) + n;
     
    return summ;
}
 
// Driver code
let n = 3;
 
document.write(sum_star_num(n));
 
// This code is contributed by rishavmahato348.
</script>

                    

Output
51

Time complexity: O(1).
Auxiliary Space: O(1)
 



Last Updated : 25 Jul, 2022
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