Find the position nth element of a rectangular tiled spiral?

Question

What is an algorithm to get the nth element of a rectangular tiled spiral?

Here is n:

[ 20  ][ 21  ][ 22  ][ 23  ][ 24  ]
[ 19  ][  6  ][  7  ][  8  ][  9  ]
[ 18  ][  5  ][  0  ][  1  ][ 10  ]
[ 17  ][  4  ][  3  ][  2  ][ 11  ]
[ 16  ][ 15  ][ 14  ][ 13  ][ 12  ]

and here are the corresponding coordinates for n:

[-2,2 ][-1,2 ][ 0,2 ][ 1,2 ][ 2,2 ]
[-2,1 ][-1,1 ][ 0,1 ][ 1,1 ][ 2,1 ]
[-2,0 ][-1,0 ][ 0,0 ][ 1,0 ][ 2,0 ]
[-2,-1][-1,-1][ 0,-1][ 1,-1][ 2,-1]
[-2,-2][-1,-2][ 0,-2][ 1,-2][ 2,-2]

If given n, how to calculate the coordinates?

Answer 1

Here is an code in JavaScript. It calculates position for 2D matrix starting with number 1 in a middle (0, 0)

13  12  11  10  25
14   3   2   9  24
15   4   1   8  23
16   5   6   7  22
17  18  19  20  21

/**
 * Finds coordinates (position) of the number
 * 
 * @param {Number} n - number to find position/coordinates for
 * @return {Number[]} - x and y coordinates of the number
 */
function position(n) {
    const k = Math.ceil((Math.sqrt(n) - 1) / 2);
    let t = 2 * k + 1;
    let m = Math.pow(t, 2);

    t -= 1;

    if (n >= m - t) {
        return [k - (m - n), -k];
    }

    m -= t;

    if (n >= m - t) {
        return [-k, -k + (m - n)];
    }

    m -= t;

    if (n >= m - t) {
        return [-k + (m - n), k];
    }

    return [k, k - (m - n - t)];
}