Baird, Paul; Fardoun, Ali; Hanna, Zeina Ghazo Quadratic cyclic sequences. (English) Zbl 1478.11033 Monatsh. Math. 193, No. 2, 195-232 (2020). Summary: We explore relations between cyclic sequences determined by a quadratic difference relation, cyclotomic polynomials, Eulerian digraphs and walks in the plane. These walks correspond to closed paths for which at each step one must turn either left or right through a fixed angle. In the case when this angle is \(2 \pi / n\), then non-symmetric phenomena occurs for \(n \ge 12\). Examples arise from algebraic integers of modulus one which are not \(n\)’th roots of unity. MSC: 11B83 Special sequences and polynomials 52C05 Lattices and convex bodies in \(2\) dimensions (aspects of discrete geometry) Keywords:cyclic sequence; quadratic difference relation; cyclotomic polynomial; digraph; Eulerian digraph; planar lattice; planar walk PDFBibTeX XMLCite \textit{P. Baird} et al., Monatsh. Math. 193, No. 2, 195--232 (2020; Zbl 1478.11033) Full Text: DOI arXiv References: [1] Baird, P., An invariance property for frameworks in Euclidean space, Linear Algebra Appl., 440, 243-265 (2014) · Zbl 1292.05166 [2] Bertrand, J., Solutions d’un problème, CRAS, 105, 369 (1887) · JFM 19.0200.03 [3] Chartrand, G.; Lesniak, L.; Zhang, P., Graphs and Digraphs (2015), New York: Chapman and Hall, New York [4] Eastwood, MG; Penrose, R., Drawing with complex numbers, Math. Intell., 22, 8-13 (2000) · Zbl 1052.51505 [5] Gessel, IM; Zeilberger, D., Random walk in a Weyl chamber, Proc. Am. Math. Soc., 115, 27-31 (1992) · Zbl 0792.05148 [6] Ghazo Hanna, Z.: Combinatorial and Geometric Cycles, Thesis, Université de Brest - Université Libanaise (2020) [7] Kelly, A.; Nagy, B., Reactive nonholonomic trajectory generation via parametric optimal control, Int. J. Robot. Res., 22, 583-601 (2003) [8] Lawler, GF; Limic, V., Random Walk: A Modern Introduction. Cambridge Studies in Advanced Mathematics (2010), Cambridge: Cambridge University Press, Cambridge · Zbl 1210.60002 [9] Looker, J.R.:Constant speed interpolating paths, DSTO Defence Science and Technology Organisation, AR 014-939. DSTO-TN-0989, March (2011) [10] Nagell, T., Introduction to Number Theory (1951), New York: Wiley, New York · Zbl 0042.26702 [11] Ran, D.; Cochran, JE Jr, Path planning and state estimation for unmanned aerial vehicles in hostile environments, J. Guid. Control Dyn., 33, 2, 595-601 (2010) This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.