Math Memes

The second triangle is 5 on account of the black triangle on the inside and the compound triangle made up of all three smaller triangles and the fourth negative space triangle. I believe the formula for how many triangles is linear because each iteration of the fractal can be represented as scooping more negative space triangles from the existing set of triangles. Each iteration you scoop out the same number of black triangles as you had white triangles the previous iteration, creating two more white triangles for every white triangle you had before, and adding one more compound triangle.

The numbers we see though from each early iteration are as follows:

1 -> 5 -> 17 -> 53 -> 161

Which happens to conform with 3(n-1)+2