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We present a general analysis of the bifurcation sequences of periodic orbits in general position of a family of reversible 1:1 resonant Hamiltonian normal forms invariant under $\Z_2\times\Z_2$ symmetry. The rich structure of these…
Horadam introduced a new generalized sequence of numbers, describing its key features and the special sub-sequences that are obtained depending on the choices of initial parameters. This sequence and its sub-sequences are known as the…
For a prime number p, we construct a generating set for the ring of invariants for the p+1 dimensional indecomposable modular representation of a cyclic group of order p^2. We then use the constructed invariants to describe the…
In this short note, we reprove in a very elementary way some known facts about Pisano periods as well as some considerations about the link between Pisano periods and the order of roots of the characteristic equation. The technics only…
We give a simple condition for a linear recurrence (mod 2^w) of degree r to have the maximal possible period 2^(w-1).(2^r-1). It follows that the period is maximal in the cases of interest for pseudo-random number generation, i.e. for…
We introduce and characterise grid classes, which are natural generalisations of other well-studied permutation classes. This characterisation allows us to give a new, short proof of the Fibonacci dichotomy: the number of permutations of…
We show that there is a bijection between the renormalizations and proper completely invariant closed sets of expanding Lorenz map, which enable us to distinguish periodic and non-periodic renormalizations. Based on the properties of…
We give multiple proofs of two formulas concerning the enumeration of permutations avoiding a monotone consecutive pattern with a certain value for the inverse peak number or inverse left peak number statistic. The enumeration in both cases…
The Fibonacci sequence $\mathbb{F}$ is the fixed point beginning with $a$ of morphism $\sigma(a,b)=(ab,a)$. Since $\mathbb{F}$ is uniformly recurrent, each factor $\omega$ appears infinite many times in the sequence which is arranged as…
In this work, we define a more general family of polynomials in several variables satisfying a linear recurrence relation. Then we provide explicit formulas and determinantal expressions. Finally, we apply these results to recurrent…
Motivated by Andrews' partitions with initial repetitions, we derive parity formulas for several functions for this class of partitions. In many cases, we present an infinite family of Ramanujan-like congruences modulo 2.
In this article we revise the problem of anomalous values of pulsars' braking indices n_{obs} and frequency second derivatives F2 arising in observations. The intrinsic evolutionary braking is buried deep under superimposed irregular…
The classical congruences satisfied by the Fibonacci and Lucas sequences are reflected with the decomposition of primes in the ring generated by the gold number. This generalizes to establish a correspondence that we hope will be new…
We consider sequences of polynomials that satisfy differential-difference recurrences. Polynomials satisfying such recurrences frequently appear as generating polynomials of integer valued random variables that are of interest in discrete…
This paper first discusses the size and orientation of hat supertiles. Fibonacci and Lucas sequences, as well as a third integer sequence linearly related to the Lucas sequence are involved. The result is then generalized to any aperiodic…
As a guiding example, the diffraction measure of a random local mixture of the two classic Fibonacci substitutions is determined and reanalysed via self-similar measures of Hutchinson type, defined by a finite family of contractions. Our…
Let $M_{d,n}(q)$ denote the number of monic irreducible polynomials in $\mathbb{F}_q[x_1, x_2, \ldots , x_n]$ of degree $d$. We show that for a fixed degree $d$, the sequence $M_{d,n}(q)$ converges $q$-adically to an explicitly determined…
In this paper, we consider the matrix polynomial obtained by using bi-periodic Fibonacci matrix polynomial. Then, we give some properties and binomial transforms of the new matrix polynomials.
In this paper we study periodicity phenomena for modular extensions between Weyl modules and between Weyl and simple modules of the general linear group that are associated to adding a power of the characteristic to the first parts of the…
Given a prime number $p$, the study of divisibility properties of a sequence $c(n)$ has two contending approaches: $p$-adic valuations and superconcongruences. The former searches for the highest power of $p$ dividing $c(n)$, for each $n$;…