Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2609.22303

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

Mike Winkler
View original source ↗Read source document ↗Export metadata ↓

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.

Explore related subjects

math.GM

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

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

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

A Proof of Liu's Conjecture on the Fundamental Triangle Inequality

Let $a,b,c$ be the side lengths of a triangle, and let $R$ and $r$ denote its circumradius and inradius, respectively. Liu proposed the conjecture \[ \sum_{\text{cyc}} \left(\frac{a(b+c-a)}{bc}\right)^k \ge 2+\left(\frac{2r}{R}\right)^k,\qquad k>1, \] with the reverse inequality for $k<1$. We prove this conjecture by reducing it to an algebraic inequality for three positive variables with prescribed sum and product. We also determine the equality cases.

math.GM↗
arXiv2026

A quadratic critical-value conjecture for the fifth Bessel moment

We conjecture an explicit evaluation of the pure fifth Bessel moment $\int_0^\infty K_0(t)^5\,dt$ as a quadratic expression in the critical value $L(f,2)$ of the weight-three, level-60 newform $f$ (LMFDB orbit 60.3.b.a) identified in the twisted fifth-moment modularity theorem of Lim, Tu and Yu, with coefficients in $\mathbb{Q}(\sqrt{5})$ and the square taken before the real and imaginary parts. Directed interval computations, using no stored Bessel or $L$-values, bound the absolute discrepancy by $10^{-358}$. We prove three exact modular identities for $f$: the Petersson-norm formula $\langle f,f\rangle_{60} = \frac{3(5-\sqrt{5})}{2π^4}|L(f,2)|^2$, the coefficient-conjugation relation $L(f^σ,2) = κL(f,2)$ with explicit $κ\in \mathbb{Q}(\sqrt{5},i)$, and the twisted symmetric-square evaluation $L(χ_{-4}\mathrm{Sym}^2 f,2) = \sqrt{15}\,π^2 \langle f,f\rangle_{60}$, together with $L(χ_{-4}\mathrm{Sym}^2 f,3) = π^4\langle f,f\rangle_{60}/8$, in the full Euler-factor normalization of Lim, Tu and Yu. The last identity shows that the companion norm conjecture $D_{5,\mathrm{odd}} = \frac{3\sqrt{15}(5-\sqrt{5})}{2}|L(f,2)|^2$ is equivalent to the symmetric-square conjecture $D_{5,\mathrm{odd}} = π^2 L(χ_{-4}\mathrm{Sym}^2 f,2)$ of Lim, Tu and Yu, while the exact relation $D_{5,\mathrm{even}} = π^2 D_{5,\mathrm{odd}}/(2\sqrt{15})$ follows from Chuang's period formulas. Every Bessel-to-modular equality, including the individual-period formula, remains conjectural. Complete proofs, exact rational certificates and verification programs are included as ancillary files.

math.GM↗