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Fibonacci along even powers is (almost) realizable

Lookup NU author(s): Professor Tom WardORCiD

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This is the authors' accepted manuscript of an article that has been published in its final definitive form by Fibonacci Association, 2022.

For re-use rights please refer to the publisher's terms and conditions.


Abstract

An integer sequence is called realizable if it is the count of periodic points of some map. The Fibonacci sequence (F_n) does not have this property, and the Fibonacci sequence sampled along the squares (F_(n^2)) also does not have this property. We prove that this is an arithmetic phenomenon related to the discriminant of the Fibonacci sequence, by showing that the sequence (5F_(n^2)) is realizable. More generally, we show that (F_(n^{2k-1})) is not realizable in a particularly strong sense while (5F_(n^{2k})) is realizable, for any k greater than or equal to 1.


Publication metadata

Author(s): Moss P, Ward T

Publication type: Article

Publication status: Published

Journal: The Fibonacci Quarterly

Year: 2022

Volume: 60

Issue: 1

Pages: 40-47

Online publication date: 01/02/2022

Acceptance date: 11/01/2021

Date deposited: 13/05/2021

ISSN (print): 0015-0517

Publisher: Fibonacci Association

URL: https://www.fq.math.ca/60-1.html


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