# 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
```

## Recommended: Please try your approach on {IDE} first, before moving on to the solution.

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++ implementation of the approach ` `#include ` `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; ` `} `

 `// 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 `

 `# 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 `

 `// 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 `

 ` `

Output:
```56
```

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