Year: 2026
Type: preprint
Abstract: Let $q\geq3$ be odd and put $m_r=\lfloor r\log_2q\rfloor$. We count parity sequences with exactly $r$ entries equal to $1$ for which the coefficient in the $qx+1$ iteration first drops below $1$ at step $m_r+1$. Their number $a_q(r)$ also counts the corresponding residue classes modulo $2^{m_r+1}$. By classifying increasing integer sequences according to the first index at which they exceed a prescribed bound, we obtain a recurrence which, for $q=3$, establishes the five- or six-term block pattern conjectured in 2015, allowing a shorter final block. We establish the sharp inequalities \[ \frac1r\binom{m_r-1}{r-1}\leq a_q(r)\leq\frac1r\binom{m_r}{r-1} \] Equality in the lower (respectively upper) bound occurs exactly when $\{r\log_2q\}$ is smaller (respectively larger) than all preceding fractional parts $\{j\log_2q\}$, $1\leq j
Source: arXiv (Cornell University)
Author: Mike Winkler
Language: English
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7212997959