Parametrizing a square spiral can be done by modifying the equations found in this OEIS entry
$$ k(n) = \frac{\pi}{2}\left \lfloor \sqrt{4n-3} \right \rfloor $$ $$ x(n) = \sum_{k=1}^{n} \sin(k(n)) $$ $$ y(n) = \sum_{k=1}^{n} \cos(k(n)) $$
However this assumes a square, rectangular spirals see below, are not covered:
5x3:
12 4 3 2 11
13 5 0 1 10
14 6 7 8 9
4x5:
23 22 21 20
15 14 13 12
4 3 2 11
5 0 1 10
6 7 8 9
16 17 18 19
Is it possible to parameterize the positions of an arbitrary rectangular spiral with width $w$ and height $h$ in a similar way?
For example in the 5x3 case shown above for each integer $n$ $x(x)$ would give: 0, 1, 1, 0, -1, -1, -1, 0, 1, 2, 2, 2, -2, -2, -2. and $y(n)$ would give 0, 0, -1,-1, -1, 0, 1, 1, 1, 1, 0, -1, -1, 0, 1.