Pattern of Triangles & Holes in Sierpinski Gasket

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Pattern of Triangles & Holes in Sierpinski GasketWe can figure the Math behind the pattern. We use two variables to track what is happening at any given step (𝑛): 𝑻_𝒏 : The number of Remaining triangles 𝑯_𝒏 : The number of Holes created Now, Step 0: 𝑇_0=1 (Just the starting triangle), 𝐻_0=0 (No holes yet) Step 1: 𝑇_1=3 (We kept 3 out of 4), 𝐻_1=1 (We removed 1) Step 2: 𝑇_2=3 × 3=9(3^2 ),𝐻_2=1(old)+3(new)=4 Step 3: 𝑇_3=3 × 3 × 3=27(3^3 ),𝐻_3=1+3+3^2= 13 The Formulas Thus, Remaining Triangles: 𝑇_𝑛=3^𝑛 Holes: 𝐻_𝑛=𝐻_(𝑛−1)+𝑇_(𝑛−1) (which expands to 1+3+├ 3^2+…+3^(𝑛−1) )