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Well, it just so happens that I figured this out for whoever was wondering...
When n is even, and n+1 is odd (obviously)
Sn+1 = Sn + (n+2)(3n+2)/4
When n is odd, and n+1 is even (obviously)
Sn+1 = Sn + (n+1)(3n+5)/4
These formulas are for the total triangles in something that looks like this...
/\
/_\
/\ /\
/_\ /_\
/\ /\ /\
/_\ /_\ /_\
This triangle has n=3 and has 13 total triangles.
The general formula for total triangles is hard to find. And because this question doesn't ask for the general solution for any n, I won't post it. It is really fun to find though...
The solution is shown below...
Total Number of triangles = (n*n) + ( (n) * (n-1) * (4n+1) / 2 ) where n = (k+1) /2. K is the number of triangles on the each side.
In the above example k is 3.
k = 1 -> Total Triangles = 1
k = 2 -> Total Triangles = 4.875 ~ 5
k = 3 -> Total Triangles = 13
k = 4 -> Total Triangles = 26.875 ~ 27
k = 5 -> Total Triangles = 48
k = 6 -> Total Triangles = 77.875 ~ 78
and so on....
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