[math-fun] A Not-So-Random Walk

In preparation for next month's tau-day I've been playing with the continued fraction thereof. Consider the sign of nth convergent of 2*pi divided by the nth convergent of pi, minus 2. The first sixty are: 0, 1, -1, 1, -1, 1, -1, 0, 1, 0, 0, 0, 0, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 0, 0, 1, 0, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, 0, 0, 0, -1, 1, -1, 1, -1. I wasn't particularly surprised that stretches of non-zeros alternated sign. What impressed me was that the alternation continued *in step* after arbitrary-length stretches of zeros.