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 Observation I 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 pattern broke down around the 8th term (of the Fibonacci sequence). mjx-math { display: inline-block; text-align: left; line-height: 0; text-indent: 0; font-style: normal; font-weight: normal; font-size: 100%; font-size-adjust: none; letter-spacing: normal; border-collapse: collapse; word-wrap: normal; word-spacing: normal; white-space: nowrap; direction: ltr; padding: 1px 0; } mjx-container[jax="CHTML"][display="true"] { display: block; text-align: center; margin: 1em 0; } mjx-container[jax="CHTML"][display="true"][width="full"] { display: flex; } mjx-container[jax="CHTML"][display="true"] mjx-math { padding: 0; } mjx-container[jax="CHTML"][justify="left"] { text-align: left; } mjx-container[jax="CHTML"][justify="right"] { text-align: right; } mjx-munder { display: inline-block; text-align: left; } mjx-over { text-align: left; } mjx-munder:not([limits="false"]) { display: inline-table; } mjx-munder > mjx-row { text-align: left; } mjx-under { padding-bottom: .1em; } mjx-TeXAtom { display: inline-block; text-align: left; } mjx-mn { display: inline-block; text-align: left; } mjx-c { display: inline-block; } mjx-utext { display: inline-block; padding: .75em 0 .2em 0; } mjx-mo { display: inline-block; text-align: left; } mjx-stretchy-h { display: inline-table; width: 100%; } mjx-stretchy-h > * { display: table-cell; width: 0; } mjx-stretchy-h > * > mjx-c { display: inline-block; transform: scalex(1.0000001); } mjx-stretchy-h > * > mjx-c::before { display: inline-block; width: init