Fibonacci Structure in Harmonic Series Partitions

A few years ago when I was in high school I was playing around with the harmonic series and noticed some correspondences to the Fibonacci sequence. My original observation was cute but the math it led to was genuinely satisfying.The ObservationI was playing around with the harmonic series and started greedily grouping terms such that their sum exceeded some threshold τ. I found that when I set this threshold to 0.5, the number of terms in each group followed the Fibonacci sequence, though this p...