Fractal Iteration Area

In this figure, it expands 3 times with each iterations(The blue area is the default).

Then use geometric series to solve:

$\begin{aligned} A&=\dfrac{1}{4}+\dfrac{3}{2^5}+\dfrac{3}{2^6}+\dfrac{3}{2^7}+\dots\\ &=\dfrac{1}{4}+\overset{\infty}{\underset{n=5}{\sum}}\dfrac{3}{2^n}\\ &=\dfrac{1}{4}+\dfrac{\dfrac{3}{32}}{1-\dfrac{1}{2}}\\ &=\blue{\dfrac{7}{16}} \end{aligned}$

Therefore, the answer is $7+16=\boxed{23}$ .

After infinitely many iterations, the final colored and uncolored (white) areas is as shown above. Let the large square be a unit square. Then the final white area is $\left(\dfrac 34\right)^2$ and the final colored area is $1-\left(\dfrac 34\right)^2 = 1 - \dfrac 9{16} = \dfrac 7{16}$ . Therefore $a+b = 7 + 16 = \boxed{23}$ .

The first iteration has $4$ squares, and after many iterations, the following $6$ pink triangles will be added:

The total area is then $4 \cdot (\frac{1}{4})^2 + 6 \cdot \frac{1}{2} \cdot (\frac{1}{4})^2 = \frac{7}{16}$ , so $a = 7$ , $b = 16$ , and $a + b = \boxed{23}$ .