A Digital Binomial Theorem

In this article, we demonstrate how the Binomial Theorem in turn arises from a one-parameter generalization of the Sierpinski triangle. The connection between them is given by the sum-of-digits function, s(k), defined as the sum of the digits in the binary representation of k (see [1]). For example, s(3) = s(1·2+1·2) = 2. Towards this end, we begin with a well-known matrix formulation of Sierpinski’s triangle that demonstrates its fractal nature (see [5], p.246). Define a sequence of matrices Sn of size 2 n × 2 recursively by