Number of points in a Bresenham's circle

Question

The Midpoint circle algorithm generates a set of quantized coordinates for a circle of a given radius. The number of points generated for is of course a multiple of $4$ due to symmetry, but I didn't succeed in writing a closed formula for the number of pixels generated.

$\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad$

For the first couple of radii (starting with $3$), the number of pixels genereated is: $$16, 20, 28, 32, 40, 44, 52, 56, 60, 68, 72...$$

So, is there a general closed form for this series?

Edit:

The series looks like it's given by: $$a_n = 4\bigg\lceil\sqrt{2}\left(n-\frac{1}{2}\right)\bigg\rceil$$ Though I can't see how this can be proved.

Answer 1

Stupid me.

Since by definition up to the quadrant midpoint, the number of pixels is a half quadrant is just: $$r\sin 45^\circ = \bigg\lfloor \frac{n-1/2}{\sqrt{2}}\bigg\rfloor$$ Multiply by 8, and we get the result..