Related papers: On the Orthogonality of Generalized Pattern Sequen…
The ordinal patterns of a fixed number of consecutive values in a time series is the spatial ordering of these values. Counting how often a specific ordinal pattern occurs in a time series provides important insights into the properties of…
In this paper, we investigate the generalized Pell sequence, the generalized complex Pell sequence and the generalized dual Pell sequence using the Pell numbers. We obtain special cases of these sequences. Furthermore, we give recurrence…
Scattered sequences are a generalization of scattered polynomials. So far, only scattered sequences of order one and two have been constructed. In this paper an infinite family of scattered sequences of order three is obtained. Equivalence…
We study pattern densities in binary sequences, finding optimal limit sequences with fixed pattern densities.
A composition of a nonnegative integer (n) is a sequence of positive integers whose sum is (n). A composition is palindromic if it is unchanged when its terms are read in reverse order. We provide a generating function for the number of…
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 consider representations of integers as sums of at most four distinct $m$-gonal numbers (allowing a fixed number of repeats of each polygonal number occurring in the sum). We show that the number of such representations…
Let $(X_{k})_{k\geq 1}$ and $(Y_k)_{k\geq 1}$ be the sequence of $X$ and $Y$-coordinates of the positive integer solutions $(x, y)$ of the equation $x^2 - dy^2 = t$. In this paper we completely describe those recurrence sequences such that…
In two recent articles we have examined a generalization of the binomial distribution associated with a sequence of positive numbers, involving asymmetric expressions of probabilities that break the symmetry {\it win-loss}. We present in…
In the paper, I consider appearance of unit's digits in minor totals of a few integer sequences. The sequences include the sequence of even integers, sequence of odd integers and Faulhaber polynomial at $p = 2$. Application of difference…
Sequences whose terms are equal to the number of functions with specified properties are considered. Properties are based on the notion of derangements in a more general sense. Several sequences which generalize the standard notion of…
An integer of the form $p_m(x)= \frac{(m-2)x^2-(m-4)x}{2} \ (m\ge 3)$, for some integer $x$ is called a generalized polygonal number of order $m$. A ternary sum $\Phi_{i,j,k}^{a,b,c}(x,y,z)=ap_{i+2}(x)+bp_{j+2}(y)+cp_{k+2}(z)$ of…
Given an integer partition of $n$ into distinct parts, the sum of the reciprocal parts is an example of an egyptian fraction. We study this statistic under the uniform measure on distinct parts partitions of $n$ and prove that, as $n \to…
We study convergence properties of sparse averages of partial sums of Fourier series of continuous functions. By sparse averages, we are considering an increasing sequences of integers $n_0 < n_1 < n_2 < ...$ and looking at…
An integer of the form $P_8(x)=3x^2-2x$ for some integer $x$ is called a generalized octagonal number. A quaternary sum $\Phi_{a,b,c,d}(x,y,z,t)=aP_8(x)+bP_8(y)+cP_8(z)+dP_8(t)$ of generalized octagonal numbers is called {\it universal} if…
Expansion of real numbers is a basic research topic in number theory. Usually we expand real numbers in one given base. In this paper, we begin to systematically study expansions in multiple given bases in a reasonable way, which is a…
For each integer $m\ge3$, let $P_m(x)$ denote the generalized $m$-gonal number $\frac{(m-2)x^2-(m-4)x}{2}$ with $x\in\mathbb{Z}$. Given positive integers $a,b,c,k$ and an odd prime number $p$ with $p\nmid c$, we employ the theory of ternary…
Good integers introduced in 1997 form an interesting family of integers that has been continuously studied due to their rich number theoretical properties and wide applications. In this paper, we have focused on classes of $2^\beta$-good…
Prime number multiplet classifications and patterns are extended to negative integers. The extension from prime numbers to single prime powers is also studied. Prime number septets at equal distance are given. It is also shown that each…
Given an increasing sequence of integers a(n), it is known (due to Weyl) that for almost all reals t, the fractional parts of the dilated sequence t*a(n) are uniformly distributed in the unit interval. Some effort has been made recently to…