What causes this apparent pattern in hailstone sequences?

Question

If $e_n$ represents the set of even numbers in the hailstone sequence starting with $n$ and $o_n$ represents the set of odd numbers in the hailstone sequence starting with $n$, plotting $\displaystyle \sum_{o \in o_n}\log_2\left({3o+1} \over {o} \right)$ in relation to $\mathbf{card}(e_n)$ gives an interesting plot, which seems to fall along a line:

Why does this relation appear?

Answer 1

First thoughts, but this needs to be investigated: $\sum \log_2 (\frac{3o+1}{o}) = \log_2 \prod (3+\frac{1}{o}) \simeq log_2 (3^{card(o_n)}+\delta_1) \simeq card(o_n) \log_23+\delta_2$

$\simeq 1,5849 card(o_n)+\delta_2$.

If you apply the collatz function to a number $n_0$ until it reaches some other $n_1$, you have something like this: $n_1=\frac{3^{card(o_n)}}{2^{card(e_n)}} n_0 + \frac{3^{card(o_n)-1}}{2^{card(e_n)}} + \frac{3^{card(o_n)-2}}{2^{card(e_n)-1-\ldots}}+\ldots+\frac{3^0}{2^{\ldots}} $

The thing is that if $n_1$ is a number near $n_0$ (e.g. you start with $n_0=63$ and you stop at $n_1=61$, i think you have a $card(o_n)=34$ and $card(e_n)=54$ in that case), you'll see that

$card(e_n)\simeq card(o_n) log_23\simeq 1,5849 card(o_n)$. Indeed, the first fraction $\frac{3^{card(o_n)}}{2^{card(e_n)}} \simeq 1$

So dividing your sum with $card(e_n)$ might bring something linear, i suspect that the $\delta$ which involve fractions of successive $n_x$ is in correlation with $\frac{n_1}{n_0}$, especially if you continue to apply the collatz function until $n_1=1$. I will have a look at it later....

edit:

I didn't had much time but from what i saw, my first impression was right: $\sum \log_2 (\frac{3o+1}{o}) \simeq 1,5849 card(o_n)+\delta_2$ (and $\delta_2$ does not seems to make a difference)

and when $n_1 = 1$ , $\frac{card(e_n)}{card(o_n)}$ is floating arround 1,8 which is not far from 1,5849 when $n_1$ is near $n_0$.

Why is it "not far" and why do $card(e_n)\simeq card(o_n) log_23$ when $n_1\simeq n_0$ are interesting questions, especially for the quest of cycles.

Answer 2

There is a paper on arxiv:"on the glide of 3x+1 problem". It shows that when stopping at the first $n_1$ below $n_0$, $2^{card(e_n)-1}<3^{card(o_n)}<2^{card(e_n)}$. This explains the values you found . And thank you for the warm welcome

Comment: Err... it was a sarcastic comment for thoose who deleted my previous answer. But thank you