Pascal Triangle Vs Sierpinski Triangle

Question

What is the relation between these two triangles?

I can remember an skew Sierpinski triangle in Pascal Triangle's page in wikipedia but it has been deleted?

Can anybody explain the relation between these two?

Answer 1

It's explained on Wikipedia - in short: you need to color Pascal's triangle and use a different color for even and odd numbers.

Pascal's triangle

If one takes Pascal's triangle with $2n$ rows and colors the even numbers white, and the odd numbers black, the result is an approximation to the Sierpinski triangle. More precisely, the limit as $n$ approaches infinity of this parity-colored $2n$-row Pascal triangle is the Sierpinski triangle.

Answer 2

As the other answer has alluded to, we see beautiful fractal patterns if you take Pascal's triangle mod n. In particular, for n=2, we see Sierpinski's triangle. We can see how these patterns arise as, for example, adding any two binomial coefficients which are even, then their sum must also be even.