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You are given a regular polygon with $$$n$$$ vertices labeled from $$$1$$$ to $$$n$$$ in counter-clockwise order. The triangulation of a given polygon is a set of triangles such that each vertex of each triangle is a vertex of the initial polygon, there is no pair of triangles such that their intersection has non-zero area, and the total area of all triangles is equal to the area of the given polygon. The weight of a triangulation is the sum of weigths of triangles it consists of, where the weight of a triagle is denoted as the product of labels of its vertices.
Calculate the minimum weight among all triangulations of the polygon.
The first line contains single integer $$$n$$$ ($$$3 \le n \le 500$$$) — the number of vertices in the regular polygon.
Print one integer — the minimum weight among all triangulations of the given polygon.
3
6
4
18
According to Wiki: polygon triangulation is the decomposition of a polygonal area (simple polygon) $$$P$$$ into a set of triangles, i. e., finding a set of triangles with pairwise non-intersecting interiors whose union is $$$P$$$.
In the first example the polygon is a triangle, so we don't need to cut it further, so the answer is $$$1 \cdot 2 \cdot 3 = 6$$$.
In the second example the polygon is a rectangle, so it should be divided into two triangles. It's optimal to cut it using diagonal $$$1-3$$$ so answer is $$$1 \cdot 2 \cdot 3 + 1 \cdot 3 \cdot 4 = 6 + 12 = 18$$$.
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Provedor Codeforces
Código CF1140D
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Datas 09/05/2023 09:46:02
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