# Problem 577: Counting hexagons
An equilateral triangle with integer side length \$n \\ge 3\$ is divided
into \$n\^2\$ equilateral triangles with side length 1 as shown in the
diagram below. The vertices of these triangles constitute a triangular
lattice with \$\\frac{(n+1)(n+2)} 2\$ lattice points. Let \$H(n)\$ be
the number of all regular hexagons that can be found by connecting 6 of
these points. For example, \$H(3)=1\$, \$H(6)=12\$ and \$H(20)=966\$.
Find \$\\displaystyle \\sum\_{n=3}\^{12345} H(n)\$.