## Polar polygons

### Problem 465

Published on Sunday, 30th March 2014, 05:00 am; Solved by 115; Difficulty rating: 85%The *kernel* of a polygon is defined by the set of points from which the entire polygon's boundary is visible. We define a *polar polygon* as a polygon for which the origin is __strictly__ contained inside its kernel.

For this problem, a polygon can have collinear consecutive vertices. However, a polygon still cannot have self-intersection and cannot have zero area.

For example, only the first of the following is a polar polygon (the kernels of the second, third, and fourth do not strictly contain the origin, and the fifth does not have a kernel at all):

Notice that the first polygon has three consecutive collinear vertices.

Let P(`n`) be the number of polar polygons such that the vertices (`x`, `y`) have integer coordinates whose absolute values are not greater than `n`.

Note that polygons should be counted as different if they have different set of edges, even if they enclose the same area. For example, the polygon with vertices [(0,0),(0,3),(1,1),(3,0)] is distinct from the polygon with vertices [(0,0),(0,3),(1,1),(3,0),(1,0)].

For example, P(1) = 131, P(2) = 1648531, P(3) = 1099461296175 and P(343) mod 1 000 000 007 = 937293740.

Find P(7^{13}) mod 1 000 000 007.