Related papers: nCk sequences and their difference sequences

Completely antipersistent binary time series are sequences in which every time that an $N$-bit string $\mu$ appears, the sequence is continued with a different bit than at the last occurrence of $\mu$. This dynamics is phrased in terms of a…

Continuous clocks, i.e. the clocks that measure time in a continuous manner, are regarded as an essential component of sensing technology. Precision and recurrence time are two basic features of continuous clocks. In this paper, in the…

We prove that the sequence $(N_k)_k$, where each $N_k$ is defined as the smallest positive integer $n$ for which the $n$th term $g_{k,n}$ of the $k$-G\"obel sequence is not an integer, is unbounded.

A $d$-subsequence of a sequence $\varphi = x_1\dots x_n$ is a subsequence $x_i x_{i+d} x_{i+2d} \dots$, for any positive integer $d$ and any $i$, $1 \le i \le n$. A \textit{$k$-Thue sequence} is a sequence in which every $d$-subsequence,…

An algorith to count, or alternatively generate, all k-element transversals of a set system is presented and compared with three known methods. For special cases it works in output-linear time.

The linear complexity of a sequence $s$ is one of the measures of its predictability. It represents the smallest degree of a linear recursion which the sequence satisfies. There are several algorithms to find the linear complexity of a…

Conventional neural architectures for sequential data present important limitations. Recurrent networks suffer from exploding and vanishing gradients, small effective memory horizons, and must be trained sequentially. Convolutional networks…

A universal cycle for permutations is a word of length n! such that each of the n! possible relative orders of n distinct integers occurs as a cyclic interval of the word. We show how to construct such a universal cycle in which only n+1…

The sequence a_1,...,a_m is a common subsequence in the set of permutations S = {p_1,...,p_k} on [n] if it is a subsequence of p_i(1),...,p_i(n) and p_j(1),...,p_j(n) for some distinct p_i, p_j in S. Recently, Beame and Huynh-Ngoc (2008)…

We determine the probability distribution for relative projective objects in an exceptional sequence of type $A_n$ of any length. We show that these events (the $j$-th object in an exceptional sequence of length $k\le n$ being relatively…

Let $\{U_n\}_{n \geqslant 0}$ and $\{G_m\}_{m \geqslant 0}$ be two linear recurrence sequences defined over the integers. We establish an asymptotic formula for the number of integers $c$ in the range $[-x, x]$ which can be represented as…

A space is `n-arc connected' (n-ac) if any family of no more than n-points are contained in an arc. For graphs the following are equivalent: (i) 7-ac, (ii) n-ac for all n, (iii) continuous injective image of a closed sub-interval of the…

A k-composition of n is a sequence of length k of positive integers summing up to n. In this paper, we investigate the number of k-compositions of n satisfying two natural coprimality conditions. Namely, we first give an exact asymptotic…

Time series motif discovery has been a fundamental task to identify meaningful repeated patterns in time series. Recently, time series chains were introduced as an expansion of time series motifs to identify the continuous evolving patterns…

We study the properties of a sequence cn defined by the recursive relation \[\frac{c_0}{n + 1}+\frac{c_1}{n + 2}+\ldots+\frac{c_n}{2n + 1}=0\] for $n>1$ and $c_0=1$. This sequence also has an alternative definition in terms of certain norm…

For each element of certain families of integer sequences, we study the term-wise ratios of the Hankel transforms of three sequences related to that element by series reversion. In each case, the ratios define well-known sequences, and in…

Let k be a positive integer. A sequence s over an n-element alphabet A is called a k-radius sequence if every two symbols from A occur in s at distance of at most k. Let f_k(n) denote the length of a shortest k-radius sequence over A. We…

Recurrent neural networks (RNNs) are capable of learning features and long term dependencies from sequential and time-series data. The RNNs have a stack of non-linear units where at least one connection between units forms a directed cycle.…

We suggest a novel method of clustering and exploratory analysis of temporal event sequences data (also known as categorical time series) based on three-dimensional data grid models. A data set of temporal event sequences can be represented…

Universal cycle for $k$-permutations is a cyclic arrangement in which each $k$-permutation appears exactly once as $k$ consecutive elements. Enumeration problem of universal cycles for $k$-permutations is discussed and one new enumerating…