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In a previous question: (http://mathoverflow.net/questions/109515/the-gauss-circle-problem-on-a-hexagonal-lattice) I asked for an analytic approximation for the number of lattice points in or along the contour of a circle centered on a lattice point in an $A_2$ hexagonal lattice. The user emiliocba noted that such an approximation was provided by:

Lax, P.D., Phillips, R.S. The asymptotic distribution of lattice points in Euclidean and non-Euclidean spaces. J. Funct. Anal. 46(3), pp. 280 – 350 (1982).

With a best current error term of $O(r^{\frac{2}{3}})$ provided by:

Levitan, B.M. Asymptotic formulae for the number of lattice points in Euclidean and Lobachevskii spaces. Russian Mathematical Surveys 42(3), pp. 13 - 42 (1987).

My question is now if there exists an exact counting solution for the number of points within a circle of radius $r$ centered on a lattice point in an $A_2$ hexagonal lattice. We know that an exact counting solution exists for the $Z^2$ integer lattice using the Floor[] function (see - http://mathworld.wolfram.com/GausssCircleProblem.html ):

$N(r) = 1 + 4*Floor[r] + 4*\sum^{Floor[r]}_{i=1} Floor[(r^2-i^2)^{\frac{1}{2}}]$

Can we write a similar counting function for the $A_2$ lattice?

For a list of example values, the number of lattice points in a circle of diameter $r$ (i.e. radius $\frac{r}{2}$) centered on a lattice point in an $A_2$ hexagonal lattice, is given in (http://oeis.org/A053416/list).

For $n = {0, 1, 2, ...}$ we have $N(r) = {1, 1, 7, 7, 19, 19, 37, 43, 61, 73, 91}$.

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# An exact counting solution for the number of points within a circle of radius $r$ centered on a lattice point in a $A_2$ hexagonal lattice

In a previous question: (http://mathoverflow.net/questions/109515/the-gauss-circle-problem-on-a-hexagonal-lattice) I asked for an analytic approximation for the number of lattice points in or along the contour of a circle centered on a lattice point in an $A_2$ hexagonal lattice. The user emiliocba noted that such an approximation was provided by:

Lax, P.D., Phillips, R.S. The asymptotic distribution of lattice points in Euclidean and non-Euclidean spaces. J. Funct. Anal. 46(3), pp. 280 – 350 (1982).

With a best current error term of $O(r^{\frac{2}{3}})$ provided by:

Levitan, B.M. Asymptotic formulae for the number of lattice points in Euclidean and Lobachevskii spaces. Russian Mathematical Surveys 42(3), pp. 13 - 42 (1987).

My question is now if there exists an exact counting solution for the number of points within a circle of radius $r$ centered on a lattice point in an $A_2$ hexagonal lattice. We know that an exact counting solution exists for the $Z^2$ integer lattice using the Floor[] function (see - http://mathworld.wolfram.com/GausssCircleProblem.html ):

$N(r) = 1 + 4*Floor[r] + 4*\sum^{Floor[r]}_{i=1} Floor[(r^2-i^2)^{\frac{1}{2}}]$

Can we write a similar counting function for the $A_2$ lattice?