Related papers: 2-Free Tetranacci Sequences
Let $p_n$ be $n$th prime, and let $(S_n)_{n=1}^\infty:=(S_n)$ be the sequence of the sums of the first $2n$ consecutive primes, that is, $S_n=\sum_{k=1}^{2n}p_k$ with $n=1,2,\ldots$. Heuristic arguments supported by the corresponding…
We consider the problem of drawing samples from posterior distributions formed under a Dirichlet prior and a truncated multinomial likelihood, by which we mean a Multinomial likelihood function where we condition on one or more counts being…
Let $\theta $ be a Salem number and $P(x)$ a polynomial with integer coefficients. It is well-known that the sequence $(\theta^n)$ modulo 1 is dense but not uniformly distributed. In this article we discuss the sequence $(P(\theta^n))$…
We study algorithmic randomness and monotone complexity on product of the set of infinite binary sequences. We explore the following problems: monotone complexity on product space, Lambalgen's theorem for correlated probability,…
We describe a simple method that produces automatically closed forms for the coefficients of continued fractions expansions of a large number of special functions. The function is specified by a non-linear differential equation and initial…
Rich words are characterized by containing the maximum possible number of distinct palindromes. Several characteristic properties of rich words have been studied; yet the analysis of repetitions in rich words still involves some interesting…
We consider a family of integer sequences generated by nonlinear recurrences of the second order, which have the curious property that the terms of the sequence, and integer multiples of the ratios of successive terms (which are also…
Latent factor model estimation typically relies on either using domain knowledge to manually pick several observed covariates as factor proxies, or purely conducting multivariate analysis such as principal component analysis. However, the…
This thesis is devoted to studying estimates of the least common multiple of some integer sequences. Our study focuses on effective bounding of the $\mathrm{lcm}$ of some class of quadratic sequences, as well as arithmetic progressions and…
Let $\mathcal{A}'$ be the set of integers missing any three fixed digits from their decimal expansion. We produce primes in a thin sequence by proving an asymptotic formula for counting primes of the form $p = m^2 + \ell^2$, with $\ell \in…
In this paper we study some structure properties of primitive weird numbers in terms of their factorization. We give sufficient conditions to ensure that a positive integer is weird. Two algorithms for generating weird numbers having a…
This paper establishes for the first time the predictive performance of speed priors and their computational complexity. A speed prior is essentially a probability distribution that puts low probability on strings that are not efficiently…
Let (F_n^{(k)})_{n\geq -(k-2)} be the k-generalized Fibonacci sequence, defined as the linear recurrence sequence whose first k terms are \(0, 0, \ldots, 0, 1\), and whose subsequent terms are determined by the sum of the preceding k terms.…
A common problem in phylogenetics is to try to infer a species phylogeny from gene trees. We consider different variants of this problem. The first variant, called Unrestricted Minimal Episodes Inference, aims at inferring a species tree…
The document tries to put focus on sequences with certain properties and periods leading to the first value smaller than the starting value in the Collatz problem. With the idea that, if all starting numbers lead ultimately to a smaller…
A Ducci sequence is a sequence of integer $n$-tuples obtained by iterating the map \[ D : (a_1, a_2, \ldots, a_n) \mapsto \big(|a_1-a_2|,|a_2-a_3|,\ldots,|a_n-a_1|\big). \] Such a sequence is eventually periodic and we denote by $P(n)$ the…
Partitioning a set of elements into an unknown number of mutually exclusive subsets is essential in many machine learning problems. However, assigning elements, such as samples in a dataset or neurons in a network layer, to an unknown and…
The $2$-adic complexity has been well-analyzed in the periodic case. However, we are not aware of any theoretical results on the $N$th $2$-adic complexity of any promising candidate for a pseudorandom sequence of finite length $N$ or…
Employing probabilistic techniques we compute best possible upper and lower bounds on the price of an option on one or two assets with continuous piecewise linear payoff function based on prices of simple call options of possibly distinct…
The Collatz conjecture is explored using polynomials based on a binary numeral system. It is shown that the degree of the polynomials, on average, decreases after a finite number of steps of the Collatz operation, which provides a weak…