Blog post: ajb: Fractal from Binomial Coefficients

Above is a discrete fractal generated by creating a table of zeros and ones by deciding if the binomial coefficients are even or odd. The "key" here is paint black if odd, otherwise leave light blue.

The pattern is closely related to Pascal's triangle.

The pattern clearly shows self-similarity as all fractals do.

As far as I know, this pattern was first noticed in [1].

A slight variant

Just for fun I used the same algorithm to study the pattern associated with modified binomial coefficients of the form

$latex left( begin{array}{c} (-1)^{k}n\ k end{array} right)$

Again the pattern shows lots of self-similarity.

References

[1] S. Wolfram: American Mathematical Monthly, 91 (November 1984) 566-571 Read and comment on the full post