Related papers: Enumerating the rationals from left to right
The existence of apparently coincidental equalities (also called Wilf-equivalences) between the enumeration sequences, or generating functions, of various hereditary classes of combinatorial structures has attracted significant interest. We…
The distribution of a given sequence in the set of all sequences with n ones and m = M - n zeros are found by relating the problem to the partitions of a natural number in m natural summands, taking into account the order. The formulas…
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…
Based on the structure of Fibonacci sequence, we give a new proof for the irrationality exponents of the Fibonacci real numbers. Moreover, we obtain all the irrationality exponents of the real numbers corresponding to the differences of…
A real number is called left-computable if there exists a computable increasing sequence of rational numbers converging to it. In this article we are investigating a proper subset of the left-computable numbers. We say that a real number…
We consider a certain linear recursive relation with integer parameters and study some of its algebraic and geometric properties, with the purpose of estimating the number of chains of valences in the Farey series.
A special class of orthogonal rational functions (ORFs) is presented in this paper. Starting with a sequence of ORFs and the corresponding rational functions of the second kind, we define a new sequence as a linear combination of the…
This report details the generation and use of tree node ordering keys in a single relational database table. The keys for each node are calculated from the keys of its parent, in such a way that the sort order places every node in the tree…
We recast homogeneous linear recurrence sequences with fixed coefficients in terms of partial Bell polynomials, and use their properties to obtain various combinatorial identities and multifold convolution formulas. Our approach relies on a…
Given a vector function ${\bf F}=(F_1,\ldots,F_d),$ analytic on a neighborhood of some compact subset $E$ of the complex plane with simply connected complement, we define a sequence of vector rational functions with common denominator in…
We define a triangular array closely related to Stern's diatomic array and show that for a fixed integer $r\geq 1$, the sum $u_r(n)$ of the $r$th powers of the entries in row $n$ satisfy a linear recurrence with constant coefficients. The…
The following article summarizes research where theorems and their respective demonstrations are postulated based on quadratic equations with special properties given by the Pythagorean triplets and the Fibonacci sequence given the second…
We discuss the use of matrices for providing sequences of rationals that approximate algebraic irrationalities. In particular, we study the regular representation of algebraic extensions, proving that ratios between two entries of the…
We create a sequence version of calculus. First, we define equivalence, some fundamental operations, differential, and integral for sequences. Then, we propose sequence versions of identity function, power function, exponential function,…
To approximate a simple root of an equation we construct families of iterative maps of higher order of convergence. These maps are based on model functions which can be written as an inner product. The main family of maps discussed is…
In this paper we study how to accelerate the convergence of the ratios (x_n) of generalized Fibonacci sequences. In particular, we provide recurrent formulas in order to generate subsequences (x_{g_n}) for every linear recurrent sequence…
We consider the question of approximating any real number $\alpha$ by sums of $n$ rational numbers $\frac{a_1}{q_1} + \frac{a_2}{q_2} + ... + \frac{a_n}{q_n}$ with denominators $1 \leq q_1, q_2, ..., q_n \leq N$. This leads to an inquiry on…
Analogously to de Bruijn sequences, Orientable sequences have application in automatic position-location applications and, until recently, studies of these sequences focused on the binary case. In recent work by Alhakim et al., recursive…
There are numerous ways to represent real numbers. We may use, e.g., Cauchy sequences, Dedekind cuts, numerical base-10 expansions, numerical base-2 expansions and continued fractions. If we work with full Turing computability, all these…
A real-valued sequence $f = \{ f(n) \}_{n \in \mathbb{N}}$ is said to be second-order holonomic if it satisfies a linear recurrence $f (n + 2) = P (n) f (n + 1) + Q (n) f (n)$ for all sufficiently large $n$, where $P, Q \in \mathbb{R}(x)$…