Any closed disk in the real plane includes a finite set of lattice points.
These roughly-circular patches of lattice points are connected by chains of
adjacent lattice points (this is an easy theorem) and hence they form a
special class of polyominoes, which I call "disc polyominoes".
It's quite easy to calculate which lattice points are within a given radius
of a given center, but the inverse problem can be a little challenging.
That is, given a polyomino, determine whether it is a disk polyomino.
I have been enumerating small disk polyominoes, to see how many
configurations are possible for various numbers of lattice points. The
resulting sequence (wake up, Neil!) is apparently not in OEIS. I make it
that there is one disk polyomino for each of the orders 0, 1, and 2; two for
each of the orders 3, 4, 5, and 6; only one for order 7; two each for orders
9 and 10; and (I'm getting less confident here) three each for orders 11 and
12. That is, the sequence is 1, 1, 1, 2, 2, 2, 2, 1, 2, 2, 3, 3 ... These
are enough entries to run off the end of OEIS.
I haven't really tried to establish any interesting results on this
sequence. It would be nice to know that it grows without limit and that it
eventually permanently exceeds any given lower bound. Two disk polyominoes
may be considered "adjacent" if one may be derived from the other by adding
or removing one lattice point. In this way all disk polyominoes form an
infinite graph; I strongly suspect this graph is connected.
Each disk polyomino has an interval of possible radii for the containing
disk; this interval is always open above, but may be either closed or open
below. The endpoints of the intervals are always square roots of rational
numbers.
