Related papers: Strongly k-recursive sequences

We revisit a technique of S. Lehr on automata and use it to prove old and new results in a simple way. We give a very simple proof of the 1986 theorem of Honkala that it is decidable whether a given k-automatic sequence is ultimately…

There is no recursively enumerable sequence of sufficiently strong 2-consistent r.e. theories such that each proves the $2$-consistency of the next. Montalb\'an and Shavrukov independently asked whether this result generalizes to…

We introduce special classes of irreducible representations of groups: thick representations and dense representations. Denseness implies thickness, and thickness implies irreducibility. We show that absolute thickness and absolute…

Deep learning techniques have been hugely successful for traditional supervised and unsupervised machine learning problems. In large part, these techniques solve continuous optimization problems. Recently however, discrete generative deep…

We define a class of sequences ${a_n}$ by $a_1=a$ and $a_{n+1}=P(a_n)$, where $P(x)$ is a polynomial with real coefficients. We then find out for which values $a$ and for which polynomials $P(x)$ these sequences will be constant after a…

A graph is called set-sequential if its vertices can be labeled with distinct nonzero vectors in $\mathbb{F}_2^n$ such that when each edge is labeled with the sum$\pmod{2}$ of its vertices, every nonzero vector in $\mathbb{F}_2^n$ is the…

A constant term sequence is a sequence of rational numbers whose $n$-th term is the constant term of $P^n(\boldsymbol{x}) Q(\boldsymbol{x})$, where $P(\boldsymbol{x})$ and $Q(\boldsymbol{x})$ are multivariate Laurent polynomials. While the…

We study the class of rational recursive sequences (ratrec) over the rational numbers. A ratrec sequence is defined via a system of sequences using mutually recursive equations of depth 1, where the next values are computed as rational…

The pure $O$-sequences of the form $(1,a,a,\ldots)$ are classified.

We discuss an interesting sequence defined recursively; namely, sequence A105774 from the On-Line Encyclopedia of Integer Sequences, and study some of its properties. Our main tools are Fibonacci representation, finite automata, and the…

We prove that every connected strongly regular graph on sufficiently many vertices is Hamiltonian. We prove this by showing that, apart from three families, connected strongly regular graphs are (highly) pseudo-random. Our results suggest a…

D-finite functions and P-recursive sequences are defined in terms of linear differential and recurrence equations with polynomial coefficients. In this paper, we introduce a class of numbers closely related to D-finite functions and…

Let L be an infinite regular language on a totally ordered alphabet (A,<). Feeding a finite deterministic automaton (with output) with the words of L enumerated lexicographically with respect to < leads to an infinite sequence over the…

In this paper, the branches of recursive and recurrent neural networks are classified in detail according to the network structure, training objective function and learning algorithm implementation. They are roughly divided into three…

Confidence sequences are anytime-valid analogues of classical confidence intervals that do not suffer from multiplicity issues under optional continuation of the data collection. As in classical statistics, asymptotic confidence sequences…

Recently, many structured tensors are defined and their properties are discussed in the literature. In this paper, we introduce a new class of structured tensors, called exceptionally regular tensor, which is relevant to the tensor…

The degree sequence of the algebraic numbers in an algebraic linear recurrence sequence is shown to be virtually periodic. This is proved using the Skolem-Mahler-Lech theorem. It has applications to the degree sequence and the minimal…

In a companion paper it was shown that the class of constant-depth determinate k-ary recursive clauses is efficiently learnable. In this paper we present negative results showing that any natural generalization of this class is hard to…

For a class of stationary regularly varying and weakly dependent time series, we prove the so-called complete convergence result for the corresponding space-time point processes. As an application of our main theorem, we give a simple proof…

In this article, we study the relationship between notions of depth for sequences, namely, Bennett's notions of strong and weak depth, and deep $\Pi^0_1$ classes, introduced by the authors and motivated by previous work of Levin. For the…