Related papers: A Study of Second-Order Linear Recurrence Sequence…
In this paper, we clarified the relationship between continued fractions, determinants, and identities, making it easier to apply these methods systematically in other settings. In particular, we studied finite continued fractions from the…
We consider $m$-th order linear recurrences that can be thought of as generalizations of the Lucas sequence. We exploit some interplay with matrices that again can be considered generalizations of the Fibonacci matrix. We introduce the…
We derive a formula which expresses a second order cumulant whose entries are products as a sum of cumulants where the entries are single factors. This extends to the second order case the formula of Krawczyk and Speicher. We apply our…
We show that, if an integer sequence is given by a linear recurrence of constant rational coefficients, then it can be represented as the difference of two arithmetic terms with exponentiation, which do not contain any irrational constant.…
A second order polynomial sequence is of Fibonacci type (Lucas type) if its Binet formula is similar in structure to the Binet formula for the Fibonacci (Lucas) numbers. In this paper we generalize identities from Fibonacci numbers and…
The classical Fibonacci sequence is known to exhibit many fascinating properties. In this paper, we explore the Fibonacci sequence and integer sequences generated by second order linear recurrence relations with positive integer…
Let G be the space of generating functions of a periodic infinite order linear recurrence. In this paper we provide an explicit procedure for computing a basis of G.
We give the trace representation of a family of binary sequences derived from Euler quotients by determining the corresponding defining polynomials. Trace representation can help us producing the sequences efficiently and analyzing their…
In this paper we introduce a new sequence of polynomials, which follow the same recursive rule of the well-known Lucas-Lehmer integer sequence. We show the most important properties of this sequence, relating them to the Chebyshev…
We derive the P-finite recurrences for classes of sequences with ordinary generating function containing roots of polynomials. The focus is on establishing the D-finite differential equations such that the familiar steps of reducing their…
Deep neural networks are a family of computational models that are naturally suited to the analysis of hierarchical data such as, for instance, sequential data with the use of recurrent neural networks. In the other hand, ordinal regression…
We study a new type of sequences whose elements are defined in terms of the position, sign and magnitude of another element of the sequence. The name ultra-recursive comes from the fact that these sequences possess terms that are generated…
An integer sequence that is defined by initial values and a linear recurrence with constant integer coefficients, can be represented by the difference of two arithmetic terms containing exponentiation. All constants occuring in the term are…
Modified third-order Jacobsthal sequence is defined in this study. Some properties involving this sequence, including the Binet-style formula and the generating function are also presented.
In this paper, we count a dual set of Stirling permutations by the number of alternating runs. Properties of the generating functions, including recurrence relations, grammatical interpretations and convolution formulas are studied.
We give a simplified presentation of some results about recurrences of certain sequences of binomial sums in terms of (generalized) Fibonacci and Lucas polynomials.
Repdigits are natural numbers formed by the repetition of a single digit. In this paper, we explore the presence of repdigits in the product of consecutive balancing or Lucas-balancing numbers.
Continual learning aims to learn on non-stationary data streams without catastrophically forgetting previous knowledge. Prevalent replay-based methods address this challenge by rehearsing on a small buffer holding the seen data, for which a…
A nearly linear recurrence sequence (nlrs) is a complex sequence $(a_n)$ with the property that there exist complex numbers $A_0$,$\ldots$, $A_{d-1}$ such that the sequence $\big(a_{n+d}+A_{d-1}a_{n+d-1}+\cdots +A_0a_n\big)_{n=0}^{\infty}$…
A class of determinants is introduced. Different kind of mathematical objects, such as Fibonacci, Lucas, Tchebychev, Hermite, Laguerre, Legendre polynomials, sums and covergents are represented as determinants from this class. A closed…