Related papers: On the Gap-sum and Gap-product Sequences of Intege…
The main goal of this paper is to study the different definitions of generating sequences appearing in the literature. We present these definitions and show that under certain situations they are equivalent. We also present an example that…
In this paper, we enumerate two families of pattern-avoiding permutations: those avoiding the vincular pattern $2-41-3$, which we call semi-Baxter permutations, and those avoiding the vincular patterns $2-41-3$, $3-14-2$ and $3-41-2$, which…
We introduce hypergeometric-type sequences. They are linear combinations of interlaced hypergeometric sequences (of arbitrary interlacements). We prove that they form a subring of the ring of holonomic sequences. An interesting family of…
The Catalan numbers $C_n$ are an extremely well-studied sequence of numbers that appear as the answer to many combinatorial problems. Two generalizations of these numbers that have been studied are the Fuss-Catalan numbers and the…
We consider series of the form $$ \frac{p}{q} +\sum_{j=2}^\infty \frac{1}{x_j}, $$ where $x_1=q$ and the integer sequence $(x_n)$ satisfies a certain non-autonomous recurrence of second order, which entails that $x_n|x_{n+1}$ for $n\geq 1$.…
The notion of transducer integer sequences is considered through a series of examples. By definition, transducer integer sequences are integer sequences produced, under a suitable interpretation, by finite automata encoding tree morphisms…
Given $d \in \mathbb{N}$, we establish sum-product estimates for finite, non-empty subsets of $\mathbb{R}^d$. This is equivalent to a sum-product result for sets of diagonal matrices. In particular, let $A$ be a finite, non-empty set of $d…
D. S. Hong and P. Pongsriiam have provided a necessary and sufficient condition for the generating function for Fibonacci numbers (resp. the Lucas numbers) to be an integer value, for rational numbers. In other words, their results relate…
We expand on the remark by Andrews on the importance of infinite sums and products in combinatorics. Let $\{g_d(n)\}_{d\geq 0,n \geq 1}$ be the double sequences $\sigma_d(n)= \sum_{\ell \mid n} \ell^d$ or $\psi_d(n)= n^d$. We associate…
This article investigates integer sequences that partition the sequence into blocks of various lengths - irregular arrays. The main result of the article is explicit formulas for numbering of irregular arrays. A generalization of Cantor…
In this paper, we introduce integer sequences satisfying new congruence properties inspired by the Euler and Gauss congruences, which we call Euler-Gauss sequences. Noting that every Gauss sequence is an Euler-Gauss sequence, we compare…
We discuss the properties of the Hankel transformation of a sequence whose elements are the sums of consecutive generalized Catalan numbers and find their values in the closed form.
We prove several power series identities involving the refined generating function of interval orders, as well as the refined generating function of the self-dual interval orders. These identities may be expressed as $\sum_{n\ge…
There are several standard procedures used to create new sequences from a given sequence or from a given pair of sequences. In this paper I discuss the most popular of these procedures. For each procedure, I give a definition and provide…
In this article, we introduce and study a new integer sequence referred to as the higher order Mersenne sequence. The proposed sequence is analogous to the higher order Fibonacci numbers and closely associated with the Mersenne numbers.…
In this survey we summarize properties of pseudorandomness and non-randomness of some number-theoretic sequences and present results on their behaviour under the following measures of pseudorandomness: balance, linear complexity,…
Fibonacci sequence, generated by summing the preceding two terms, is a classical sequence renowned for its elegant properties. In this paper, leveraging properties of generalized Fibonacci sequences and formulas for consecutive sums of…
In this paper, we establish the irrationality of some open problems in mathematics based on using a recursive formula that generate the complete sequence of numbers. see [1] But before getting into that we begin with some Ramanujan notable…
Number sequences defined by a linear recursion relation are studied by means of generating functions. Indices of the terms in the recursion relation have arbitrary differenses. In addition to formulas for the nth term an algorithm is…
A generating function for reciprocal binomial coefficients is written down, integral representations of this function are obtained, generating functions for sums of reciprocal binomial coefficients are derived, new identities are obtained,…