Find the sum of the diagonal elements of the given N X N spiral matrix

Given N which is the size of the N X N spiral matrix of the form:

16 15 14 13
5  4  3  12
6  1  2  11
7  8  9  10

The task is to find the sum of the diagonal elements of this matrix.

Examples:

Input: N = 3
Output: 25
5 4 3
6 1 2
7 8 9
The sum of elements along its two diagonals will be 
1 + 3 + 7 + 5 + 9 = 25

Input: N = 5
Output: 101


Approach: Idea behind the solution is to use the concept of Dynamic Programming. We will use array dp[] to store our solution. N given in the problem can either be even or odd.
When i is odd, we have to add only 4 corner elements in dp[i – 2].

dp[i] = dp[i – 2] + (i – 2) * (i – 2) + (i – 1) + (i – 2) * (i – 2) + 2 * (i – 1) + (i – 2) * (i – 2) + 3 * (i – 1) + (i – 2) * (i – 2) + 4 * (i – 1)
dp[i] = dp[i – 2] + 4 * (i – 2) * (i – 2) + 10 * (i – 1)
dp[i] = dp[i – 2] + 4 * (i) * (i) – 6 * (i – 1)

Similarly, we can check that the above formula is true when i is even.

Below is the implementation of the above approach:

C++

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// C++ implementation of the approach
#include <bits/stdc++.h>
using namespace std;
  
// Function to return the sum of both the
// diagonal elements of the required matrix
int findSum(int n)
{
    // Array to store sum of diagonal elements
    int dp[n + 1];
  
    // Base cases
    dp[1] = 1;
    dp[0] = 0;
  
    // Computing the value of dp
    for (int i = 2; i <= n; i++) {
        dp[i] = (4 * (i * i))
                - 6 * (i - 1) + dp[i - 2];
    }
  
    return dp[n];
}
  
// Driver code
int main()
{
    int n = 4;
  
    cout << findSum(n);
  
    return 0;
}

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Java

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// Java implementation of the approach
class GFG
{
      
// Function to return the sum of both the
// diagonal elements of the required matrix
static int findSum(int n)
{
    // Array to store sum of diagonal elements
    int[] dp = new int[n + 1];
  
    // Base cases
    dp[1] = 1;
    dp[0] = 0;
  
    // Computing the value of dp
    for (int i = 2; i <= n; i++) 
    {
        dp[i] = (4 * (i * i)) - 6
                    (i - 1) + dp[i - 2];
    }
  
    return dp[n];
}
  
// Driver code
public static void main(String args[])
{
    int n = 4;
  
    System.out.println(findSum(n));
}
}
  
// This code is contributed by Akanksha Rai

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Python3

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# Python 3 implementation of the approach
  
# Function to return the sum of both the
# diagonal elements of the required matrix
def findSum(n):
      
    # Array to store sum of diagonal elements
    dp = [0 for i in range(n + 1)]
  
    # Base cases
    dp[1] = 1
    dp[0] = 0
  
    # Computing the value of dp
    for i in range(2, n + 1, 1):
        dp[i] = ((4 * (i * i)) - 6 * 
                      (i - 1) + dp[i - 2])
  
    return dp[n]
  
# Driver code
if __name__ == '__main__':
    n = 4
  
    print(findSum(n))
  
# This code is contributed by
# Surendra_Gangwar

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C#

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// C# implementation of the approach
  
class GFG
{
      
// Function to return the sum of both the
// diagonal elements of the required matrix
static int findSum(int n)
{
    // Array to store sum of diagonal elements
    int[] dp = new int[n + 1];
  
    // Base cases
    dp[1] = 1;
    dp[0] = 0;
  
    // Computing the value of dp
    for (int i = 2; i <= n; i++) 
    {
        dp[i] = (4 * (i * i))
                - 6 * (i - 1) + dp[i - 2];
    }
  
    return dp[n];
}
  
// Driver code
static void Main()
{
    int n = 4;
  
    System.Console.WriteLine(findSum(n));
}
}
  
// This code is contributed by mits

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PHP

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<?php
// PHP implementation of the approach
  
// Function to return the sum of both the
// diagonal elements of the required matrix
function findSum($n)
{
      
    // Array to store sum of diagonal elements
    $dp = array();
  
    // Base cases
    $dp[1] = 1;
    $dp[0] = 0;
  
    // Computing the value of dp
    for ($i = 2; $i <= $n; $i++) 
    {
        $dp[$i] = (4 * ($i * $i)) - 6 * 
                       ($i - 1) + $dp[$i - 2];
    }
  
    return $dp[$n];
}
  
// Driver code
$n = 4;
  
echo findSum($n);
  
// This code is contributed by Akanksha Rai
?>

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Output:

56


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