A new statistic for the $3x+1$ problem

Abstract:

A finite $(3x+1)$–trajectory is a sequence $\underline { a}=a_1,\ldots ,a_n$ of positive integers such that $a_{i+1}=3a_i+ 1$ if $a_i$ is odd, $a_{i+1}=a_i/2$ if $a_i$ is even, $a_i>1$ if $i<n$ and $a_n=1$. For such a sequence $\underline { a}$ we define $C(\underline { a}) = (a_1a_2+\cdots +a_{n-1}a_n+a_na_1)/(a_1^2+\cdots +a_n^2)$. We prove that $9/13<C(\underline { a})<5/7$ if $a_1$ is odd and $a_1\ge 3$. Histograms suggest that $C$ may have an interesting limiting distribution.