Shadow Sequences of Integers: From Fibonacci to Markov and Back

• Published in: The Mathematical intelligencer Vol. 45; no. 1; pp. 50 - 54
• Main Author: Ovsienko, Valentin
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.03.2023; Springer Verlag
• Subjects: Mathematical and Computational Engineering; Mathematical and Computational Physics; Mathematical Gems and Curiosities; Mathematical Methods in Physics; Mathematics; Mathematics and Statistics; Numerical and Computational Physics; Simulation; Theoretical
• ISSN: 0343-6993; 1866-7414
• DOI: 10.1007/s00283-021-10154-x

When Sophie Morier-Genoud and I took over the editorship of the Gems and Curiosities column from Sergei Tabachnikov a year ago, we understood that there was a long tradition of very large shoes to fill. In particular, in its previous incarnation under the name Mathematical Entertainments, this column enjoyed its golden years under the watch of David Gale. For an account, see the amazing book [4]. Integer sequences was one of the main subjects discussed in the column at that time [3]. The sequences now called Gale-Robinson and Somos sequences first appeared in the column. Gale's "riddles" about integer sequences strongly influenced combinatorics; see for instance the powerful and now classical reference [2], in which Gale's riddles were solved. Although any attempt to come up to the high bar set by David Gale on the subject of integer sequences is doomed to fail, I will discuss here a large class of integer sequences. The following general idea looks crazy. What if another integer sequence follows each integer sequence like a shadow ? I will demonstrate that this is indeed the case, perhaps not for every integer sequence (this unfortunately I don't know), but for many of them. Does this mean that the number 341962 of known and registered sequences (cf. [6]) will double? The answer is "no" for two reasons: shadows of known sequences can also be known, and one sequence can have several shadows.