2016/09/19 by Shalosh B. Ekhad, N. J. A. Sloane, Ekhad, Shalosh B. +3
Computer Science · Mathematics · #Advanced Combinatorial Mathematics #Algorithms and Data Compression #Combinatorics (math.CO) #FOS: Mathematics #Mathematical Dynamics and Fractals #Number Theory (math.NT) #math.CO #math.NT
paper · pdf · doi:10.48550/arxiv.1609.05570
10 pages, accompanied by a Maple package and six outputs files available from http://www.math.rutgers.edu/~zeilberg/mamarim/mamarimhtml/pisot.html Exclusively published in the authors' websites and this arxiv.org . [This version corrects a typo. The correct A078608 is replaced by the previous erroneous A-number.]
openalex publication_date 2016/09/19 · arxiv created 2016/09/27 · arxiv updated 2016/09/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Pisot sequences (sequences an with initial terms a0=x, a1=y, and defined for n>1 by an= \lfloor an-12/an-2 + (1)/(2) \rfloor) often satisfy linear recurrences with constant coefficients that are valid for all n ≥ 0, but there are also cautionary examples where there is a linear recurrence that is valid for an initial range of values of n but fails to be satisfied beyond that point, providing further illustrations of Richard Guy's celebrated "Strong Law of Small Numbers". In this paper we present a decision algorithm, fully implemented in an accompanying Maple program (\tt Pisot.txt), that first searches for a putative linear recurrence and then decides whether or not it holds for all values of n. We also explain why the failures happen (in some cases the `fake' linear recurrence may be valid for thousands of terms). We conclude by defining, and studying, higher-order analogs of Pisot sequences, and point out that similar phenomena occur there, albeit far less frequently. This article is dedicated to Richard K. Guy (b. Sept. 30, 1916) on his 100th birthday.