A979 | Equilateral Triangles--Platinum

Farmer John's pasture can be represented by a $N\times N$ square grid $(1\le
N\le 300)$ consisting of positions $(i,j)$ for all $1\le i,j\le N$. For each
square of the grid, the corresponding character in the input is equal to '*'
if there exists a single cow at that position and '.' if there does not exist
a cow at that position.
FJ believes that the beauty of his pasture is directly proportional to the
number of triples of cows such that their positions are equidistant from each
other. In other words, they form an equilateral triangle. Unfortunately, it
was only quite recently that FJ realized that since all of his cows are
located at integer coordinates, no beautiful triples can possibly exist if
Euclidean distance is used! Thus, FJ has decided to switch to use "Manhattan"
distance instead. Formally, the Manhattan distance between two positions
$(x_0,y_0)$ and $(x_1,y_1)$ is equal to $|x_0-x_1|+|y_0-y_1|$.
Given the grid representing the positions of the cows, compute the number of
equilateral triples.

There will be fourteen test cases aside from the sample, one for each of $N\in
\\{50,75,100,125,150,175,200,225,250,275,300,300,300,300\\}.$

The first line contains a single integer $N.$
For each $1\le i\le N,$ line $i+1$ of the input contains a string of length
$N$ consisting solely of the characters '*' and '.'. The $j$th character
describes whether there exists a cow at position $(i,j)$ or not.