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Per the title, I'm seeking the definition of a function $f(n, m)$ which evaluates to the number of lines made from exactly $n$ points which can be placed on a two-dimensional discrete, square grid of size $m \times m$.

My primary interest is the case where $n = 3$, as this is the answer I'm really after. I would've raised this question alone, however since I see that $n = 2$ has been answered already, I figured I might as well just generalize the question in hopes that there's some nice, elegant solution out there.

Per the title, I'm seeking the definition of a function $f(n, m)$ which evaluates to the number of lines made from exactly $n$ points which can be placed on a two-dimensional discrete, square grid of size $m \times m$.

My primary interest is the case where $n = 3$, as this is the answer I'm really after. I would've raised this question alone, however since I see that $n = 2$ has been answered already, I figured I might as well just generalize the question in hopes that there's some nice, elegant solution out there.

Per the title, I'm seeking the definition of a function $f(n, m)$ which evaluates to the number of lines made from exactly $n$ points which can be placed on a two-dimensional discrete, square grid of size $m \times m$.

I've opened a separate question for the case where $n = 3$, as this is the answer I'm really after. I would've raised this question alone, however since I see that $n = 2$ has been answered already, I figured I might as well just generalize the question in hopes that there's some nice, elegant solution out there.

Per the title, I'm seeking the definition of a function $f(n, m)$ which evaluates to the number of lines made from exactly $n$ points which can be placed on a two-dimensional discrete, square grid of size $m \times m$.

My primary interest is the case where $n = 3$, as this is the answer I'm really after. I would've raised this question alone, however since I see that $n = 2$ has been answered already, I figured I might as well just generalize the question in hopes that there's some nice, elegant solution out there.

Per the title, I'm seeking the definition of a function $f(n, m)$ which evaluates to the number of lines made from exactly $n$ points which can be placed on a two-dimensional discrete, square grid of size $m \times m$.

As an aside, I actually only need to know the definition for $n = 3$ where $m$ is odd (not certain that matters), however since I see that $n = 2$ has been answered already, I figured I might as well generalize the question. If it turns out that answering the general form of this question is quite difficult, I will edit to only include $n = 3$.

Per the title, I'm seeking the definition of a function $f(n, m)$ which evaluates to the number of lines made from exactly $n$ points which can be placed on a two-dimensional discrete, square grid of size $m \times m$.

I've opened a separate question for the case where $n = 3$, as this is the answer I'm really after. I would've raised this question alone, however since I see that $n = 2$ has been answered already, I figured I might as well just generalize the question in hopes that there's some nice, elegant solution out there.