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Two convex lattice polygons are equivalent if there is a lattice-preserving affine transformation mapping one of them to the other. Equivalent polygons have the same area. Let $H(A)$ denote the number of equivalence classes of such polygons having area $A$. In the paper "On the number of convex lattice polygons" Barany and Pach proved that $\log H(A)\asymp A^{1/3}$. (In 1980 Arnolʹd proved that $c_1A^{1/3}<\log H(A)<c_2A^{1/3}\log A$.)

Let $H_3(A)$ denote the number of equivalence classes of lattice triangles having area $A$. What is known about behaviour of $H_3(A)$?

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  • $\begingroup$ Can you calculate $H_3(A)$ for the first few values of $A$, and then check the Online Encyclopedia of Integer Sequences? $\endgroup$ Feb 18, 2014 at 1:00
  • $\begingroup$ My calculations give the sequence 1,1,2,3,2,4,3,5,4 which is not presented in OEIS. A shorter sequence 1,1,2,3,2,4,3,5 gives a relevant reference to A054384: Number of inequivalent sublattices of index $n$ in hexagonal lattice, where two lattices are considered equivalent if one can be rotated to give the other. But it is another problem. $\endgroup$ Feb 18, 2014 at 2:20

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Let $A$ be double area of triangle. I calculate number of equivalence classes of triangles $OPQ$ with labelled vertices. Your problem is slightly different, but mine has nicer answer:) Without loss of generality two vertices are $(0,0)$ and $(a,0)$, where $a$ is some divisor of $A=ab$, and the third vertex has coordinates $(u,b)$. Unimodular transforms which fixes vertices $(0,0)$ and $(a,0)$ (and also upper half-plane) is $(x,y)\rightarrow (x+cy,y)$. That is, third vertex $(u,b)$ is equivalent to $(u+cb,b)$. Therefore there are $b$ different classes of equivalence. Sum up by pairs $(a,b)$ such that $ab=A$, we get that number of equivalence classes equals $\sigma(A)=$sumof divisors of $A$.

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  • $\begingroup$ This number is known because it is a number of sublattices in $\mathbb{Z}^2$ of index $A$. I think that I can answer my question but I'm not sure that this problem is new. $\endgroup$ Sep 27, 2015 at 5:01

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