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I want to ask if exists an exact solution for the number of points within a circle of radius r for an honeycomb lattice. I know that it is exist for an square lattice https://mathworld.wolfram.com/GausssCircleProblem.html: $$N(r)=1+4\lfloor r \rfloor+4 \sum_{x=1}^{\lfloor r \rfloor}{\left\lfloor \sqrt{r^2−x^2} \right\rfloor}$$ ($\lfloor · \rfloor$ is the floor function.) Can we write a similar counting function for the honeycomb lattice? For an honeycomb lattice like this: An honeycomb lattice.

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  • $\begingroup$ There is a closely related question about the hexagonal lattice. $\endgroup$
    – Timothy Budd
    Commented Feb 19, 2022 at 21:07
  • $\begingroup$ How is this different, Mihaela, from the question you asked earlier, mathoverflow.net/questions/416474/… (other than having less detail than that earlier question)? $\endgroup$
    – Gerry Myerson
    Commented Feb 20, 2022 at 2:30
  • $\begingroup$ This appears to be the hexagonal lattice with a sublattice removed, so the result should follow straightforwardly from the result for the hexagonal lattice. $\endgroup$
    – Peter Taylor
    Commented Feb 20, 2022 at 13:38

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