Chapter 4: Systems Based on Numbers

Section 2: Elementary Arithmetic

Digit count sequences

Starting say with {1} repeatedly replace list by

Join[list, IntegerDigits[Apply[Plus, list], 2]]

The resulting sequences grow in length roughly like n Log[n]. The picture below shows the fluctuations around m/2 of the cumulative number of 1's up to position m in the sequence obtained at step 1000. A definite nested structure similar to picture (c) on page 130 is evident.

Image Source Notebooks:

From Stephen Wolfram: A New Kind of Science [citation]