Sum of matrix in which each element is absolute difference of its row and column numbers

Given a positive integer n. Consider a matrix of n rows and n columns, in which each element contain absolute difference of its row number and numbers. The task is to calculate sum of each element of the matrix.

Examples : 

Input : n = 2
Output : 2
Matrix formed with n = 2 with given constraint:
0 1
1 0
Sum of matrix = 2.

Input : n = 3
Output : 8
Matrix formed with n = 3 with given constraint:
0 1 2
1 0 1
2 1 0
Sum of matrix = 8.

Method 1 (Brute Force): Simply construct a matrix of n rows and n columns and initialize each cell with absolute difference of its corresponding row number and column number. Now, find the sum of each cell.

Below is the implementation of above idea : 

8

Time Complexity: O(N²), as we are traversing the matrix using nested loops.
Auxiliary Space: O(N²), as we are using extra space for generating and storing the Matrix.

Method 2 (O(n)): 

Consider n = 3, matrix formed will be: 
0 1 2 
1 0 1 
2 1 0

Observe, the main diagonal is always 0 since all i are equal to j. The diagonal just above and just below will always be 1 because at each cell either i is 1 greater than j or j is 1 greater than i and so on. 
Following the pattern we can see that the total sum of all the elements in the matrix will be, for each i from 0 to n, add i*(n-i)*2. 

Below is the implementation of above idea :

8

Time Complexity: O(N), as we are only using single loop to traverse.
Auxiliary Space: O(1), as we are not using any extra space.

Method 3 (Trick): 
Consider n = 3, matrix formed will be: 
0 1 2 
1 0 1 
2 1 0
So, sum = 1 + 1 + 1 + 1 + 2 + 2. 
On Rearranging, 1 + 2 + 1 + 2 + 2 = 1 + 2 + 1 + 2². 
So, in every case we can rearrange the sum of matrix so that the answer always will be sum of first n - 1 natural number and sum of square of first n - 1 natural number. 

Sum of first n natural number = ((n)*(n + 1))/2.
Sum of first n natural number = ((n)*(n + 1)*(2*n + 1)/6.

Below is the implementation of above idea : 

8

Time Complexity: O(1), as we are not using any loops.
Auxiliary Space: O(1), as we are not using any extra space.