Admissible $qx+1$ Sequences, Semiconvergents, and Rational Catalan Numbers

Mike Winkler

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arXiv · 2609.22303

Admissible $qx+1$ Sequences, Semiconvergents, and Rational Catalan Numbers

Mike Winkler

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Abstract

For an odd integer $q\geq3$, let $a_q(r)$ count finite words in $\{q/2,1/2\}$ containing exactly $r$ factors $q/2$, with every proper prefix product greater than $1$ and total product less than $1$. For $q=3,5,7$, these are OEIS \OEIS{A100982}, \OEIS{A174795}, and \OEIS{A174796}. Put \[ α_q=\log_2q,\qquad m_r(q)=\lfloor rα_q\rfloor. \] Using the classical cycle lemma in this $q$-specific setting, we recover the two natural Beatty passage bounds \[ \frac1r\binom{m_r(q)-1}{r-1} \leq a_q(r)\leq \frac1r\binom{m_r(q)}{r-1}, \] and prove their equality criteria directly. The lower and upper equality orders are the denominators of the strict lower and upper one-sided best approximations to $\log_2q$; together they are the denominators of all convergents and semiconvergents. Equivalently, they are the strict record minima and maxima of the binary mantissas $q^r/2^{m_r(q)}$. At every nontrivial equality order, the admissible words are in explicit bijection with rational Dyck paths, so that $a_q(r)$ is a rational Catalan number. We also prove monotonicity in $q$, compare the three OEIS sequences, and derive their growth constants and exact normalized oscillations.

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BibTeX RIS

Mike Winkler. 2026-09-14. Admissible $qx+1$ Sequences, Semiconvergents, and Rational Catalan Numbers. https://arxiv.org/abs/2609.22303

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