• Corpus ID: 9287876

# 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}
}
• Published 2 July 1999
• Mathematics
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
• T. Ward
• Computer Science, Mathematics
Electron. 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
• Mathematics
Proceedings 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
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
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
• A. Windsor
• Mathematics
Ergodic 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
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
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

## References

SHOWING 1-10 OF 11 REFERENCES
An Introduction to Symbolic Dynamics and Coding
• Computer Science
• 1995
Requiring only a undergraduate knowledge of linear algebra, this first general textbook includes over 500 exercises that explore symbolic dynamics as a method to study general dynamical systems.
The spectra of nonnegative integer matrices via formal power series
• Mathematics
• 2000
An old problem in matrix theory is to determine the n-tuples of complex numbers which can occur as the spectrum of a matrix with nonnegative entries (see [BP94, Chapter 4] or [Min88, Chapter VII]).
The new book of prime number records
1 How Many Prime Numbers Are There?.- I. Euclid's Proof.- II. Goldbach Did It Too!.- III. Euler's Proof.- IV. Thue's Proof.- V. Three Forgotten Proofs.- A. Perott's Proof.- B. Auric's Proof.- C.
An Introduction to the Theory of Numbers
THIS book must be welcomed most warmly into X the select class of Oxford books on pure mathematics which have reached a second edition. It obviously appeals to a large class of mathematical readers.
generatingfunctionology
• 1994
Mathematics Subject Classification. 11B39
• Mathematics Subject Classification. 11B39
• 1991
Mathematics Subject Classification. 11B39
• Mathematics Subject Classification. 11B39
• 1991
E
• M. Wright. An Introduction to the Theory of Numbers. 5th ed. Oxford: Clarendon,
• 1979