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Problem 351

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Hexagonal orchards

A hexagonal orchard of order n is a triangular lattice made up of points within a regular hexagon with side n. The following is an example of a hexagonal orchard of order 5:

Highlighted in green are the points which are hidden from the center by a point closer to it. It can be seen that for a hexagonal orchard of order 5, 30 points are hidden from the center.

Let H(n) be the number of points hidden from the center in a hexagonal orchard of order n.

H(5) = 30. H(10) = 138. H(1 000) = 1177848.

Find H(100 000 000).

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六边形果园

n阶六边形果园是在边长为n的正六边形中的三角形网格。如下所示是5阶六边形果园：

站在中间向外看，图中用绿色标出的点都会被更近的点挡住。可以看出，对于5阶六边形果园，从中间向外看时有30个点会被挡住。

对于n阶六边形果园，记从中间向外看时会被挡住的点的数目为H(n)。

已知H(5) = 30，H(10) = 138，H(1 000) = 1177848。

求H(100 000 000).

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