2550. Count Collisions of Monkeys on a Polygon

Medium

138

431

There is a regular convex polygon with `n`

vertices. The vertices are labeled from `0`

to `n - 1`

in a clockwise direction, and each vertex has **exactly one monkey**. The following figure shows a convex polygon of `6`

vertices.

Each monkey moves simultaneously to a neighboring vertex. A neighboring vertex for a vertex `i`

can be:

- the vertex
`(i + 1) % n`

in the clockwise direction, or - the vertex
`(i - 1 + n) % n`

in the counter-clockwise direction.

A **collision** happens if at least two monkeys reside on the same vertex after the movement or intersect on an edge.

Return *the number of ways the monkeys can move so that at least one collision*

`10`^{9 }+ 7

.**Note** that each monkey can only move once.

**Example 1:**

Input:n = 3Output:6Explanation:There are 8 total possible movements. Two ways such that they collide at some point are: - Monkey 1 moves in a clockwise direction; monkey 2 moves in an anticlockwise direction; monkey 3 moves in a clockwise direction. Monkeys 1 and 2 collide. - Monkey 1 moves in an anticlockwise direction; monkey 2 moves in an anticlockwise direction; monkey 3 moves in a clockwise direction. Monkeys 1 and 3 collide. It can be shown 6 total movements result in a collision.

**Example 2:**

Input:n = 4Output:14Explanation:It can be shown that there are 14 ways for the monkeys to collide.

**Constraints:**

`3 <= n <= 10`

^{9}

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