Get diagonal values of spiral matrix

Question

I have a n*n spiral matrix.

if N = 4

then matrix :

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

if N = 3

 7 8 9
 6 1 2
 5 4 3 

I want to get the diagonal values of this spiral matrix.

In the n=4 case diagonal values would be 7,1,3,13,10,2,4,16

I can do this by storing this matrix in array and traversing for each diagonal value.
Is there any better way to get these values.

Answer 1

To get the numbers on the main diagonal, we can notice that the values are

1 = 1
1 + 2 = 3
1 + 2 + 4 = 7
1 + 2 + 4 + 6 = 13

So the general formula is 1 + (sum i = 0 to k of 2*i) for k = 0, 1, 2, ... Simplifying this, we get k^2 + k + 1 for k = 0, 1, 2, ...

In PHP, we can generate these via something like this:

function mainDiagonal($n) {
    $values = array();

    for ($k = 0; $k < $n; $k++) {
        $values[] = $k*$k + $k + 1;
    }

    return $values;
}

To get the numbers on the antidiagonal for even N we see:

2 = 2
2 + 2 = 4
2 + 2 + 6 = 10
2 + 2 + 6 + 6 = 16

If we continue this pattern for larger matrices we see the general formula is

sum i = 0 to k of floor(i/2)*4 + 2 for k = 0, 1, 2, ...

Similarly for odd N we find the formula is

1 + (sum i = 0 to k of ceil(i/2)*4) for k = 0, 1, 2, ...

In PHP, we can generate these via something like this:

function antiDiagonal($n) {
    $values = array();

    if ($n % 2 == 0) {
        for ($k = 0; $k < $n; $k++) {
            $accum = 0;

            for ($j = 0; $j <= $k; $j++) {
                $accum += floor($j/2)*4 + 2;
            }

            $values[] = $accum;
        }
    } else {
        for ($k = 0; $k < $n; $k++) {
            $accum = 1;

            for ($j = 0; $j <= $k; $j++) {
                $accum += ceil($j/2)*4;
            }

            $values[] = $accum;
        }
    }

    return $values;
}

Notice that the maximum value of k is one less than the dimension of the matrix.

Combining these functions, we obtain:

array_unique(array_merge(mainDiagonal($n), antiDiagonal($n)))