A beautiful geometric sequence!
The Sierpinski Triangle involves a sequence of geometric figures. The first figure in the sequence is an equilateral. The second has an inverted (white) equilateral inscribed inside an equilateral triangle as shown. Each subsequent figure in this sequence is obtained by inserting an inverted (white) triangle inside each non-inverted (shaded) triangle of the previous figure. How many regions ( both white and shaded together) are in the
tenth
figure in this sequence?
For example, the first three figures in the sequence have 1 region, 4 regions and 13 regions respectively.
The answer is 29524.
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Sierpinski triangle
The number of regions is given by the recurrence R ( 1 ) = 1 , R ( 2 ) = 4 and R ( n ) = 3 R ( n − 1 ) + 1 for n > 1 . The value of R ( 1 0 ) can then be obtained by solving this recurrence or using the formula R ( n ) = 2 3 n − 1 . The number of regions turns out to be 2 9 5 2 4 .