Admissible qx+1 Sequences, Semiconvergents, and Rational Catalan Numbers
Abstract
Let q≥3 be odd and put mr=⌊ rlog2q⌋. 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 mr+1. Their number aq(r) also counts the corresponding residue classes modulo 2mr+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 1/rbinommr−1r−1≤ aq(r)≤1/rbinommrr−1 Equality in the lower (respectively upper) bound occurs exactly when {rlog2q} is smaller (respectively larger) than all preceding fractional parts {jlog2q}, 1≤ j<r. These indices are characterized by denominators of convergents and semiconvergents from below and above; at these indices r≥2, the values aq(r) are rational Catalan numbers. We also characterize the dependence on q and prove, for each fixed odd q≥3, that aq(r)=Θq(r−3/2Bqr), where Bq=(log2q)log2q/(log2q−1)log2q−1. For q=3, this proves Hikawa's conjectured polynomial factor for the number of finite binary words of given weight for which the length of every nonempty prefix is smaller than log2 3 times the number of its ones.
Related
BibTeX
@misc{winkler2026admissible,
title = {{Admissible $qx+1$ Sequences, Semiconvergents, and Rational Catalan Numbers}},
author = {Mike Winkler},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.22303},
eprint = {2609.22303},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.22303},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Admissible qx+1 Sequences, Semiconvergents, and Rational Catalan Numbers
AU - Mike Winkler
PY - 2026
DA - 2026/10/18
PB - arXiv
DO - 10.48550/arxiv.2609.22303
UR - https://arxiv.org/abs/2609.22303
AB - Let q≥3 be odd and put m_r=⌊ rlog_2q⌋. 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 1/rbinomm_r−1r−1≤ a_q(r)≤1/rbinomm_rr−1 Equality in the lower (respectively upper) bound occurs exactly when {rlog_2q} is smaller (respectively larger) than all preceding fractional parts {jlog_2q}, 1≤ j<r. These indices are characterized by denominators of convergents and semiconvergents from below and above; at these indices r≥2, the values a_q(r) are rational Catalan numbers. We also characterize the dependence on q and prove, for each fixed odd q≥3, that a_q(r)=Θ_q(r^−3/2B_q^r), where B_q=(log_2q)^log_2q/(log_2q−1)^log_2q−1. For q=3, this proves Hikawa's conjectured polynomial factor for the number of finite binary words of given weight for which the length of every nonempty prefix is smaller than log_2 3 times the number of its ones.
ER -