相关论文: The Ultimate Signs of Second-Order Holonomic Seque…

An infinite sequence $\langle{u_n}\rangle_{n\in\mathbb{N}}$ of real numbers is holonomic (also known as P-recursive or P-finite) if it satisfies a linear recurrence relation with polynomial coefficients. Such a sequence is said to be…

We study log-concavity properties of real sequences $(a_n)_{n \ge 0}$ satisfying a $d$-th order linear recurrence whose coefficients are linear functions of $n$; the so-called P-recursive (or holonomic) sequences. Writing the recurrence in…

The Hofstadter Q-sequence, with its simple definition, has defied all attempts at analyzing its behavior. Defined by a simple nested recurrence and an initial condition, the sequence looks approximately linear, though with a lot of noise.…

In this paper, we study the periodicity structure of finite field linear recurring sequences whose period is not necessarily maximal and determine necessary and sufficient conditions for the characteristic polynomial~\(f\) to have exactly…

In this paper, given a simple linear recurrence sequence of algebraic numbers, which has either a dominant characteristic root or exactly two characteristic roots of maximal modulus, we give some explicit lower bounds for the index beyond…

We consider two algorithms which can be used for proving positivity of sequences that are defined by a linear recurrence equation with polynomial coefficients (P-finite sequences). Both algorithms have in common that while they do succeed…

We investigate the conditions on an integer sequence f(n), n 2 N, with f(1) = 0, such that the sequence q(n), computed recursively via q(n) = q(n - q(n - 1)) + f(n), with q(1) = 1, exists. We prove that f(n + 1) - f(n) in {0,1}, n > 0, is a…

Let f(t) be a rational function of degree at least 2 with rational coefficients. For a given rational number x_0, define x_{n+1}=f(x_n) for each nonnegative integer n. If this sequence is not eventually periodic, then the difference…

A sequence $f_n(q)$ is $q$-holonomic if it satisfies a nontrivial linear recurrence with coefficients polynomials in $q$ and $q^n$. Our main theorems state that $q$-holonomicity is preserved under twisting, i.e., replacing $q$ by $\omega q$…

Given a holonomic sequence $F(n)$, we characterize rational functions $r(n)$ so that $r(n)F(n)$ can be summable. We provide upper and lower bounds on the degree of the numerator of $r(k)$ and show the denominator of $r(n)$ can be read from…

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…

We consider real polynomials in one variable without vanishing coefficients and with all roots real and of distinct moduli. We show that the signs of the coefficients define the order of the moduli of the roots on the real positive…

Let $G=(\mathbb Z/n\mathbb Z) \oplus (\mathbb Z/n\mathbb Z)$. Let $\mathsf {s}_{\leq k}(G)$ be the smallest integer $\ell$ such that every sequence of $\ell$ terms from $G$, with repetition allowed, has a nonempty zero-sum subsequence with…

We continue work started in [1] concerning integer sequences q(n), n in N, defined by q(n) = q(n-q(n-1)) + f(n), with q(1) = 1. Here, f(n), with f(1) = 0, is a given sequence. We define F as the set of semi-infinite sequence f such that the…

The refined inertia of a square real matrix $A$ is the ordered $4$-tuple $(n_+, n_-, n_z, 2n_p)$, where $n_+$ (resp., $n_-$) is the number of eigenvalues of $A$ with positive (resp., negative) real part, $n_z$ is the number of zero…

We study the parity-perturbed Hofstadter-type recursion $$ Q(1)=Q(2)=1,\qquad Q(n)=Q(n-Q(n-1))+Q(n-Q(n-2))+(-1)^n . $$ We prove, by computer-certified finite-state induction, that this recursion is well-defined for all $(n\ge 1)$. The proof…

Let $(G_n(x))_{n=0}^\infty$ be a $d$-th order linear recurrence sequence having polynomial characteristic roots, one of which has degree strictly greater than the others. Moreover, let $m\geq 2$ be a given integer. We ask for…

It was recently conjectured that every component of a discrete-time rational dynamical system is a solution to an algebraic difference equation that is linear in its highest-shift term (a quasi-linear equation). We prove that the conjecture…

Let $\epsilon\in \{-1,1\}$. A sequence of prime numbers $p_1, p_2, p_3, ...$, such that $p_i=2p_{i-1}+\epsilon$ for all $i$, is called a {\it Cunningham chain} of the first or second kind, depending on whether $\epsilon =1$ or -1…

Non-linear recurrences which generate integers in a surprising way have been studied by many people. Typically people study recurrences that are linear in the highest order term. In this paper I consider what happens when the recurrence is…