Equilateral Triangles

Farmer John's pasture is an $N\times N$ square grid $(1\le N\le 300)$ with positions $(i,j)$ for all $1\le i,j\le N$. A cell contains '*' if a single cow stands at that position, and '.' if no cow is there.

FJ believes the beauty of his pasture is directly proportional to the number of triples of cows whose positions are equidistant from each other, that is, triples that form an equilateral triangle. All cows are at integer coordinates, so no such triple can exist under Euclidean distance. FJ therefore switches to Manhattan distance. The Manhattan distance between $(x_0,y_0)$ and $(x_1,y_1)$ is $|x_0-x_1|+|y_0-y_1|$.

Given the grid, count the equilateral triples.

The first line contains a single integer $N$. For each $1\le i\le N$, line $i+1$ contains a string of length $N$ made of the characters '*' and '.'. The $j$th character describes whether there is a cow at position $(i,j)$.

Output a single integer, the answer. The answer fits into a signed 32-bit integer.

There are three cows, and they form an equilateral triple because the Manhattan distance between each pair of cows is equal to two.