What is the equation to find the number of unique triangles in each subdivision of the Sierpinski triangle?

Answer 1

To start we have 1 triangle (call this n(0) )
Then we add 3 more, n(1) = 1+3 = 4
Then 12 more, for n(2) = 4 + 12 = 16
etc.
Each iteration will add 4 times the previous number we added, or
n(x+1) = 1 + 3 + 4*3 + 4^2*3 + 4^3*3 + ... + 4^(x-1)*3
= 1 + 3[1 + 4 + 4^2 + ... + 4^(x-1)]
= 1 + 4^x - 1
or n(x) = 4^x

Mad Mac has raised a point of symmantic ambiguity in the question. All triangles after the nth iteration are "unique". But if you were in fact referring to similarly-sized triangles (i.e., all triangles produced in the nth iteration (which are of unique size) and not all triangles in the construction after the nth-iteration) then the answer is:
4^n - 4^(n-1) = 3*4^(n-1). As the Sierpinski triangle doesn't have "colors" (nor can I put my hands on his edition of that book right now) I can neither confirm nor deny what he is talking about; however, there is no subset of triangles which share common dimensions or iteration level and with cardinality of 3^n or 3*n with the exception of n=1.

Answer 2

I am not a mathematician but have access to math books. The previous answer by "scott r" gave the total number of triangles whereas you asked for the number of unique triangles. I assume the number of unique triangles are those colored black if n =0 is black and new triangles are white. The number of black triangles after "n" iterations (subdivisions?) is therefore Nn = 3*n. The number of white triangles would be 4*n - 3*n.