Problem 292

Pythagorean Polygons

We shall define a pythagorean polygon to be a convex polygon with the following properties:

there are at least three vertices,
no three vertices are aligned,
each vertex has integer coordinates,
each edge has integer length.

For a given integer n, define P(n) as the number of distinct pythagorean polygons for which the perimeter is ≤ n.
Pythagorean polygons should be considered distinct as long as none is a translation of another.

You are given that P(4) = 1, P(30) = 3655 and P(60) = 891045.
Find P(120).

毕达哥拉斯多边形

我们定义毕达哥拉斯多边形为满足下列性质的凸多边形：

有至少三个顶点，
不存在三个顶点共线，
每个顶点坐标均为整数，
每条边长度均为整数。

对于给定的整数n，记P(n)为周长≤ n的不同毕达哥拉斯多边形的数目。
只要不能将其中一个通过平移得到另一个，就被认为是不同的毕达哥拉斯多边形。

已知P(4) = 1，P(30) = 3655，以及P(60) = 891045。
求P(120)。