Related papers: Criteria for apwenian sequences

Let $n$ be a nonnegative integer. The $n$-th Ap\'{e}ry number is defined by $$ A_n:=\sum_{k=0}^n\binom{n+k}{k}^2\binom{n}{k}^2. $$ Z.-W. Sun ever investigated the congruence properties of Ap\'{e}ry numbers and posed some conjectures. For…

In [18] we have shown that, for $p_{1},p_{2}\in(2,\infty]$, the constants of Bennett's inequality on unimodular bilinear forms on $\ell_{p_{1}}^{n_{1} }\times\ell_{p_{2}}^{n_{2}}$ are asymptotically bounded by $1$. In the present paper we…

We look at the number $L(n)$ of $O$-sequences of length $n$. Recall that an $O$-sequence can be defined algebraically as the Hilbert function of a standard graded $k$-algebra, or combinatorially as the $f$-vector of a multicomplex. The…

In this note further clue decisive observations on cobweb admissible sequences are shared with the audience. In particular an announced proof of the Theorem 1 (by Dziemia\'nczuk) from [1] announced in India -Kolkata- December 2007 is…

We give a $q$-congruence whose specializations $q=-1$ and $q=1$ correspond to supercongruences (B.2) and (H.2) on Van Hamme's 1997 list: $$ \sum_{k=0}^{(p-1)/2}(-1)^k(4k+1)A_k\equiv p(-1)^{(p-1)/2}\pmod{p^3} \quad\text{and}\quad…

We study the monic polynomials orthogonal with respect to a symmetric perturbed Gaussian weight $$ w(x;t):=\mathrm{e}^{-x^2}\left(1+t\: x^2\right)^\lambda,\qquad x\in \mathbb{R}, $$ where $t> 0,\;\lambda\in \mathbb{R}$. This weight is…

We compute the asymptotics of the determinants of certain $n\times n$ Toeplitz + Hankel matrices $T_n(a)+H_n(b)$ as $n\to\infty$ with symbols of Fisher-Hartwig type. More specifically we consider the case where $a$ has zeros and poles and…

We describe new, simple, recursive methods of construction for orientable sequences over an arbitrary finite alphabet, i.e. periodic sequences in which any sub-sequence of n consecutive elements occurs at most once in a period in either…

In their 1938 seminal paper on symbolic dynamics, Morse and Hedlund proved that every aperiodic infinite word $x\in A^N,$ over a non empty finite alphabet $A,$ contains at least $n+1$ distinct factors of each length $n.$ They further showed…

In many circumstances, given an ordered sequence of one or more types of elements or symbols, the objective is to determine the existence of any randomness in the occurrence of one specific element, say type 1. This method can help detect…

Martin Aigner introduced Catalan-like numbers as elements of the first column of admissible matrices and studied Hankel determinants of their forward shifts. In this paper we collect some properties of the Hankel determinants of the other…

English: We continue the study of G. Castelnuovo on the group of birational transformation of the complex plane that fix each point of a curve of genus >1; we use adjoint linear system of the curve as Castelnuovo does. We prove that these…

Let $E/\mathbb Q$ be an elliptic curve, and denote by $N(p)$ the number of $\mathbb{F}_p$-points of the reduction modulo $p$ of $E$. A conjecture of Koblitz, refined by Zywina, states that the number of primes $p \leq X$ at which $N(p)$ is…

We establish asymptotic formulas for the determinants of finite Toeplitz + Hankel matrices of size N, as N goes to infinity for singular generating functions defined on the unit circle in the special case where the generating function is…

Let $R$ be a commutative noetherian ring and $f_{1}, ..., f_{r} \in R$. In this article we give (cf. the Theorem in \S2) a criterion for $f_{1}, ..., f_{r}$ to be regular sequence for a finitely generated module over $R$ which strengthens…

For a certain family of complete modular lattices, we prove a Jordan--H\"older--Scheier-like" theorem with no assumptions on cardinality or well-orderedness. This family includes both lattices which are both join- and meet-continuous, as…

In this paper we study combinatorial aspects of permutations of $\{1,\ldots,n\}$ and related topics. In particular, we prove that there is a unique permutation $\pi$ of $\{1,\ldots,n\}$ such that all the numbers $k+\pi(k)$ ($k=1,\ldots,n$)…

One of the most interesting results of the last century was the proof completed by Matijasevich that computably enumerable sets are precisely the diophantine sets [MRDP Theorem, 9], thus settling, based on previously developed machinery,…

In the late 1980s, Abrusci, Girard and van de Wiele defined a variant of Goodstein sequences: the so-called inverse Goodstein sequence. In their work, they show that it terminates precisely at the Bachmann-Howard ordinal. This reveals that…

For a sequence $S$ over a finite abelian group, let $MZ(S)$ denote the length of the shortest nonempty zero-sum subsequence of $S$. We prove that if $G$ is finite abelian of order $n$ and $S$ has length $n$, then $MZ(S)\le n-|\supp(S)|+1$.…