Chapter 4: Systems Based on Numbers

Section 5: Mathematical Constants

Runs of digits [in numbers]

One can consider any base 2 digit sequence as consisting of successive runs of 0's and 1's, constructed from the list of run lengths by

Fold[Join[#1, Table[1 - Last[#1], {#2}]] &, {0}, list]

This representation is related to so-called surreal numbers (though with the first few digits different). The number with run lengths corresponding to successive integers (so that the nth digit is Mod[Floor[1/2 + Sqrt[2n]], 2]) turns out to be (1 - 21/4 EllipticTheta[2, 0, 1/2] + EllipticTheta[3, 0, 1/2])/2, and appears at least not to be algebraic.

Image Source Notebooks:

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