Related papers: Periodicity in Rectangular Arrays

A recurrence relation is said to have the Laurent property if all of its iterates are Laurent polynomials in the initial values with integer coefficients. We consider a family of nonlinear recurrences with the Laurent property, which were…

We investigate a variant of the octahedron recurrence which lives in a 3-dimensional lattice contained in [0,n] x [0,m] x R. Generalizing results of David Speyer math.CO/0402452, we give an explicit non-recursive formula for the values of…

Here, we give upper and lower bounds on the count of positive integers $n\le x$ dividing the $n$th term of a nondegenerate linearly recurrent sequence with simple roots.

We solve an asymptotic problem in the geometry of numbers, where we count the number of singular $n\times n$ matrices where row vectors are primitive and of length at most T. Without the constraint of primitivity, the problem was solved by…

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…

A two-dimensional string is simply a two-dimensional array. We continue the study of the combinatorial properties of repetitions in such strings over the binary alphabet, namely the number of distinct tandems, distinct quartics, and runs.…

We generalize the familiar notion of periodicity in sequences to a new kind of pseudoperiodicity, and we prove some basic results about it. We revisit the results of a 2012 paper of Shevelev and reprove his results in a simpler and more…

In this paper, we analyze the periodic factors of Sturmian words for the findings to lead to a linear-time algorithm for the computation of runs in this class of words which, to our best knowledge, is an open problem in literature.

A linear recurrence sequence in a cyclotomic field produces a sequence of the generating fields of each term. We show that the later sequence is periodic after removing the first finite terms, and give a bound of its period. This can be…

We consider the problem of finding repetitive structures and inherent patterns in a given string $\s{s}$ of length $n$ over a finite totally ordered alphabet. A border $\s{u}$ of a string $\s{s}$ is both a prefix and a suffix of $\s{s}$…

In this paper, we prove: Let A be a nonnegative primitive tensor with order m and dimension n. Then its primitive degree R(A)\leq (n-1)^2+1, and the upper bound is sharp. This confirms a conjecture of Shao [7].

In [A. Frid, S. Puzynina, L.Q. Zamboni, \textit{On palindromic factorization of words}, Adv. in Appl. Math. 50 (2013), 737-748], it was conjectured that any infinite word whose palindromic lengths of factors are bounded is ultimately…

Let $k$ be a field and let $R$ be a countable dimensional prime von Neumann regular $k$-algebra. We show that $R$ is primitive, answering a special case of a question of Kaplansky.

We study the number of primes with a given primitive root and in an arithmetic progression under the assumption of a suitable form of the generalized Riemann Hypothesis. Previous work of Lenstra, Moree and Stevenhagen has given asymptotics…

The evaluation of iterated primitives of powers of logarithms is expressed in closed form. The expressions contain polynomials with coefficients given in terms of the harmonic numbers and their generalizations. The logconcavity of these…

For a class of Lucas sequences ${x_n}$, we show that if $n$ is a positive integer then $x_n$ has a primitive prime factor which divides $x_n$ to an odd power, except perhaps when $n = 1, 2, 3 or 6$. This has several desirable consequences.

The article describes the structural and algorithmic relations between Cartesian trees and Lyndon Trees. This leads to a uniform presentation of the Lyndon table of a word corresponding to the Next Nearest Smaller table of a sequence of…

We study primitive divisors of terms of the sequence P_n=n^2+b, for a fixed integer b which is not a negative square. It seems likely that the number of terms with a primitive divisor has a natural density. This seems to be a difficult…

We consider Riordan arrays $\bigl(1/(1-t^{d+1}), ~ tp(t)\bigr)$. These are infinite lower triangular matrices determined by the formal power series $1/(1-t^{d+1})$ and a polynomial $p(t)$ of degree $d$. Columns of such matrix are eventually…

We consider dynamical systems arising from substitutions over a finite alphabet. We prove that such a system is linearly repetitive if and only if it is minimal. Based on this characterization we extend various results from primitive…