# A DYNAMICAL PROPERTY UNIQUE TO THE LUCAS SEQUENCE FIBONNACI QUARTERLY

@inproceedings{Puri1999ADP, title={A DYNAMICAL PROPERTY UNIQUE TO THE LUCAS SEQUENCE FIBONNACI QUARTERLY}, author={Yash Puri and Thomas Ward}, year={1999} }

The only recurrence sequence satisfying the Fibonacci recurrence and realizable as the number of periodic points of a map is (a multiple of) the Lucas sequence.

## 19 Citations

INTEGER SEQUENCES AND PERIODIC POINTS

- Mathematics
- 2002

Arithmetic properties of integer sequences counting periodic points are studied, and applied to the case of linear recurrence sequences, Bernoulli numerators, and Bernoulli denominators.

Integer sequences and dynamics

- Computer Science, MathematicsElectron. Notes Discret. Math.
- 2018

There are hints of a quite widespread Polya–Carlson dichotomy in algebraic and geometric settings, and one of these connections occurs between prime numbers and closed orbits.

Time-changes preserving zeta functions

- MathematicsProceedings of the American Mathematical Society
- 2019

We associate to any dynamical system with finitely many periodic orbits of each length a collection of possible time-changes of the sequence of periodic point counts that preserve the property of…

THE FIBONACCI QUARTERLY

- 2010

By Zeckendorf’s theorem, an equivalent definition of the Fibonacci sequence (appropriately normalized) is that it is the unique sequence of increasing integers such that every positive number can be…

Primes in divisibility sequences

- Mathematics
- 2001

We give an overview of two important families of divisibility sequences: the Lehmer--Pierce family (which generalise the Mersenne sequence) and the elliptic divisibility sequences. Recent…

On multiplicatively dependent linear numeration systems, and periodic points

- Mathematics, Computer ScienceRAIRO Theor. Informatics Appl.
- 2002

A sequence of integers which is equal to the number of periodic points of a sofic dynamical system associated with some Parry number is defined.

Smoothness is not an obstruction to realizability

- MathematicsErgodic Theory and Dynamical Systems
- 2008

Abstract A sequence of non-negative integers $(\phi _n)_{n =1}^\infty $ is said to be realizable if there is a map T of a set X such that ϕn=#{x:Tnx=x}. We prove that any realizable sequence can be…

From tent-like functions to Lucas sequences

- 2005

Counting the number of periodic points of a function leads to results on congruences for the terms of a sequence. Fermat’s little theorem (and also its generalization by Euler) can be proved in a…

A combinatorial problem about binary necklaces and attractors of Boolean automata networks

- Computer Science, MathematicsArXiv
- 2016

A formal argument supporting the following idea is provided: addition of cycle intersections in network structures causes exponential reduction of the networks' number of attractors.

Fibonacci along even powers is (almost) realizable.

- Mathematics
- 2020

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…

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