On the number of antipodal bicolored necklaces
An \textit{antipodal bicolored necklace} is a congruence class modulo the dihedral group of the set of colorings of the vertices of a regular \((2n)\)-gon with two colors, such that antipodal vertices receive different colors. In [\textit{J. J. Montellano-Ballesteros} and \textit{R. Strausz}, J. Comb. Theory, Ser. A 106, No. 1, 109--121 (2004; Zbl 1042.05024)], it was proved that the number \(\nu(2n)\) of antipodal bicolored necklaces is related to the number of combinatorial types of \((n-3)\)-dimensional convex polytopes with \(n\) vertices in general position, of which there are \(\nu(2n) - \lceil {n \over 2} \rceil\). As a consequence, \(\nu(2n)\) can be explicitly computed when \(n\) is a power of 2 or when \(n\) is an odd prime. The main result of the paper under review states that the sequence \((\nu(2n))_{ n \geq 1 }\) is non-holonomic, i.e., it does not satisfy a linear recurrence of a fixed finite order with polynomial coefficients. The authors derive this result by finding the first two terms of the asymptotic expansion of~\(\nu(2n)\).
scientific article; zbMATH DE number 6008792