How many unit squares of a square grid overlap a circle of given radius centered on the origin? Consider, in the plane, the unit squares with corners having integral rectangular coordinates. Let $N_r$ be the number of these unit squares whose interior is intersected by a circle of radius $r$ centered on the origin. Counting these, I find the sequence $(N_r)_{r\in\mathbb{N}}=(0,4,12,20,28,28,44,52,60,68,68,84,92\ldots)$, or $({1\over 4}N_r)_{r\in\mathbb{N}}=(0,1,3,5,7,7,11,13,15,17,17,21,23,\ldots),$ neither of which appears in the OEIS, nor has searching turned up anything online.
Is there anything published about this sequence? Is it somehow obtainable from the known formulas for the solution of Gauss's circle problem or concerning circle lattice points (i.e., counting lattice points inside or on a circle of radius $r$)?
Here are some examples showing only the first quadrant:

Apparently, $\lim_{r\to\infty}({1\over r}N_r)=8$ (but how to prove it?):

(This is related to an older question, where a comment refers to algorithms for rasterizing a circle, but --although that turned out to be what the asker was looking for-- those algorithms don't seem relevant to the present question, as they generally seem to produce fewer than $N_r$ grid points.)
 A: This is to supplement the accepted answer by sketching a "geometrical" argument that $N_r=8r-a(r)$, where $a(r)$ is the number of lattice points on a circle of radius $r$ centered on the origin.
Here we suppose horizontal and vertical gridlines connect all the lattice points that define the corners of the unit squares, and let an "overlap square" be any one of these unit squares whose interior is intersected by the circle.
First, by inspection it is clear that the circle touches exactly $8r$ gridlines (i.e. $2r$ gridlines per quadrant), noting that to touch a lattice point is to touch two gridlines simultaneously.
Second, there is exactly one overlap cell per touch point, because one new overlap square is entered upon passing through any touch point (which may be on either one or two gridlines).
Finally, the number of gridlines touched equals the number of touch points plus the number of lattice points touched (again because to touch a lattice point is to touch two gridlines simultaneously). Thus, $8r = N_r + a(r)$, and the required result follows.
Note that $a(r)=S(r^2)$, and both are described with a variety of algorithms in OEIS:
$a(n)$ is the number lattice points on a circle of radius $n$ (OEIS #A046109): $(1, 4, 4, 4, 4, 12, 4, 4, 4, 4, 12, 4, 4, 12, 4, 12, 4, 12, 4, 4, 12, 4, 4, 4, 4, 20,...)$
$S(n)$ is the number lattice points on a circle of radius $\sqrt{n}$ (OEIS #A004018): $(1, 4, 4, 0, 4, 8, 0, 0, 4, 4, 8, 0, 0, 8, 0, 0, 4, 8, 4, 0, 8, 0, 0, 0, 0, 12, 8, 0, 0,... )$
