Suppose I have an infinite grid. How many sets of grid points are there that contain $N$ contiguous grid points, and include the grid point at the origin?
So for example, if $N$ = 2, then there are 4 possibilities on a square grid:
0#0 000 000 000
0#0 0## ##0 0#0
000 000 000 0#0
(Here I have not counted diagonal neighbors as being contiguous.) Is there a name for this quantity? Are there known formulas or tabulated values?
Given an $n$, the number you want is $$nP(n)$$ where $P(n)$ is the number of fixed polyominoes of size $n$. A fixed polyomino in your terminology is just a set of contiguous grid points without the specification that it must include the origin. The origin specification means we want to multiply the number of fixed polyominoes by $n$, because each of the $n$ points on a fixed polyomino could be the base point.
You can find a list of the number of fixed polyominoes here: OEIS A001168.