Thanks Heureka   

Here is a vey interesting you tube clip on FRACTALS

https://www.youtube.com/watch?v=XwWyTts06tU

How many triangles do we have at:

1. Step 6 ?
2. Step 8 ?
3. Step 19 ?
4. Step 50 ?

I. The "empty" triangles in the middle are included:

$$\\ \begin{array}{rcrcrcr} s_1 &=& 1 &=& 1 &=& 1\\ s_2 &=& 3s_1+1 &=& 4 &=& 1+3\\ s_3 &=& 3s_2+1 &=& 13 &=& 1+3+3^2\\ s_4 &=& 3s_3+1 &=& 40 &=& 1+3+3^2+3^3\\ s_5 &=& 3s_4+1 &=& 121 &=& 1+3+3^2+3^3+3^4\\ s_6 &=& 3s_5+1 &=& 364 &=& 1+3+3^2+3^3+3^4+3^5\\ \cdots\\ s_n &=& 3s_{n-1}+1&=& && \dfrac{3^n-1} {2} \end{array}$$

$$\\ \rm{a.)}\qquad s_6 = \dfrac{3^6-1} {2} = 364 \\\\ \rm{b.)}\qquad s_8 = \dfrac{3^8-1} {2} = 9841\\\\ \rm{c.)}\qquad s_{19} = \dfrac{3^{19}-1} {2} = 581130733\\\\ \rm{d.)}\qquad s_{50} = \dfrac{3^{50}-1} {2} = 358~ 948~ 993~ 845~ 926~ 294 ~385~124$$

II. The "empty" triangles in the middle aren't included:

$$\\ \begin{array}{rcrcrcr} s_1 &=& 1 &=& 1 &=& 3^0\\ s_2 &=& 3s_1 &=& 3 &=& 3^1\\ s_3 &=& 3s_2 &=& 9 &=& 3^2\\ s_4 &=& 3s_3 &=& 27 &=& 3^3\\ s_5 &=& 3s_4 &=& 81 &=& 3^4\\ s_6 &=& 3s_5 &=& 243 &=& 3^5\\ \cdots\\ s_n &=& 3s_{n-1}&=& && 3^{n-1} \end{array}$$

$$\\\rm{a.)}\qquad s_6 = 3^5 = 243 \\\\\rm{b.)}\qquad s_8 = 3^7 = 2187 \\\\\rm{c.)}\qquad s_{19} = 3^{18} = 387420489 \\\\\rm{d.)}\qquad s_{50} = 3^{49} = 239 ~299 ~329 ~230 ~617 ~529 ~590 ~083$$

Twice as good as I expected: You've got

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CONGRATULATIONS !    

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