EDIT: I realized I'd tricked myself by working with a too special case of $f$, the question is now updated (*boundary lattice points* replaced *vertices*).

Suppose I have a convex lattice polygon $P$, and let $i$ be the number of internal lattice points of $P$, and $v$ be the number of ~~vertices~~ boundary lattice points of $P$. Taking this $P$ as input to [some stuff I do]*, I get the number

$i^2+(3+2v)i-1$

out. *What, if anything, does this number tell me about $P$?* I would love if it were something along the lines of Pick's formula (which from the number of internal lattice points and boundary lattice points gives the Euclidean area of the polygon), but really any insight is appreciated.

*Does this look at all familiar to anyone?*

(Addendum: I would actually be interested in anything else we can derive from knowing the numbers $i$ and $v$ here, not necessarily just the particular number here.)

*Background: The "some stuff I do" is to compute a minimal Fuchsian operator $L\in \mathbb{C}[t,\nabla]$ that kills the period integral $\int_{|x|=|y|=\epsilon} \frac{dx dy}{(1-tf)xy}$ where $f$ is a Laurent polynomial with Newton polygon $P$; the mysterious number shows up as the degree $d$ of the leading term $t^d\nabla^r+\cdots$.