I am having a hard time finding results on counting the number of polygons (with integer vertices) bounded by $((0,0),(d,0),(0,d))$ for $d\in \mathbb{N}$.
[I am curious as to how many distinct Newton polytopes can exist for a bivariate polynomial of degree $d$ or less. (Having it in $n$ dimensions is the dream, but let's start small)]
For example, when $d=2$, there are already $44$ distinct non-trivial polygons (including all dimensions). ($6$ points, $\binom{6}{2}=15$ segments, and $23$ full dimensional polygons).
When $d=3$, there are $10$ points, $45$ segments, and $220$ full dimensional polytopes (total $275$).
$d=4$ is $15+105+1499=1619$
The sequence $1,7,44,275, 1619$ does not come up on OEIS.