发表机构
Faculty of Mathematics, Ruhr University Bochum（波鸿鲁尔大学数学系）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究可容许的$qx+1$序列计数，利用循环引理得到Beatty界并证明相等准则，揭示其与半收敛子及有理Catalan数的联系，同时证明单调性并分析OEIS序列。
AI 中文摘要
对于奇数整数 \\(q\geq3\\)，设 \\(a_q(r)\\) 计数包含恰好 \\(r\\) 个因子 \\(q/2\\) 的有限词 \\(\{q/2,1/2\}\\)，其中每个真前缀乘积大于 \\(1\\) 且总乘积小于 \\(1\\)。对于 \\(q=3,5,7\\)，这些分别是OEIS序列\OEIS{A100982}、\OEIS{A174795}和\OEIS{A174796}。令 \\[ \alpha_q=\log_2q,\qquad m_r(q)=\lfloor r\alpha_q\rfloor. \\] 在此 \\(q\\) 特定设置中使用经典循环引理，我们恢复了两个自然的Beatty通道界 \\[ \frac1r\binom{m_r(q)-1}{r-1} \leq a_q(r)\leq \frac1r\binom{m_r(q)}{r-1}, \\] 并直接证明了它们的相等性准则。下界和上界相等阶分别是 \\(\log_2q\\) 的严格下和严格上单侧最佳逼近的分母；它们合在一起是所有收敛子和半收敛子的分母。等价地，它们是二进制尾数 \\(q^r/2^{m_r(q)}\\) 的严格记录最小值和最大值。在每个非平凡相等阶，可容许词与有理Dyck路径存在显式双射，因此 \\(a_q(r)\\) 是有理Catalan数。我们还证明了关于 \\(q\\) 的单调性，比较了三个OEIS序列，并推导了它们的增长常数和精确归一化振荡。
英文摘要
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.
Comments14 pages, 2 tables
