Related papers: Classes of Gap Balancing Numbers

In this paper, we introduce a generalization of Balancing and Balancing-Lucas numbers. We describe some of their properties also we give the related matrix representation and divisibility properties.

Balancing numbers $n$ are originally defined as the solution of the Diophantine equation $1+2+\cdots+(n-1)=(n+1)+\cdots+(n+r)$, where $r$ is called the balancer corresponding to the balancing number $n$. By slightly modifying, $n$ is the…

A problem of bounding the generalization error of a classifier f in H, where H is a "base" class of functions (classifiers), is considered. This problem frequently occurs in computer learning, where efficient algorithms of combining simple…

In this work, we determined the general terms of all almost balancing numbers of first and second type in terms of balancing numbers and conversely we determined the general terms of all balancing numbers in terms of all almost balancing…

Comparative prime number theory is the study of the {\em{discrepancies}} of distributions when we compare the number of primes in different residue classes. This work presents a list of the problems being investigated in comparative prime…

In this work, we defined neo balcobalancing numbers, neo Lucas-balcobalancing numbers, neo balcobalancers and neo Lucas-balcobalancers and derived the general terms of these numbers in terms of balancing numbers. Conversely we deduced the…

The balancing numbers $B_n$ ($n=0,1,\cdots$) are solutions of the binary recurrence $B_n=6B_{n-1}-B_{n-2}$ ($n\ge 2$) with $B_0=0$ and $B_1=1$. In this paper we show several relations about the sums of product of two balancing numbers of…

Imbalance learning is a subfield of machine learning that focuses on learning tasks in the presence of class imbalance. Nearly all existing studies refer to class imbalance as a proportion imbalance, where the proportion of training samples…

In this paper several examples of gaps (lacunes) between dimensions of maximal and submaximal symmetric models are considered, which include investigation of number of independent linear and quadratic integrals of metrics and counting the…

Balancing and Lucas-balancing numbers are solutions of a Diophantine equation and satisfy a second order homogeneous recurrence relation. Interestingly, these numbers can be seen as numerators and denominators in the steady state…

A positive integer $n$ is called a balancing number if there exists a positive integer $r$ such that $1 + 2 + \cdots + (n-1) = (n+1) + (n+2) + \cdots + (n+r)$. The corresponding value $r$ is known as the balancer of $n$. If $n$ is a…

The Three Gap Theorem states that for any $\alpha \in \mathbb{R}$ and $N \in \mathbb{N}$, the fractional parts of $\{ 0\alpha, 1\alpha, \dots, (N - 1)\alpha \}$ partition the unit circle into gaps of at most three distinct lengths. We prove…

We pursue the study of families of functions on the natural numbers, with emphasis here on the bounded families. The situation being more complicated than the unbounded case, we attack the problem by classifying the families according to…

Positive integers with all digits equal are called repdigits. In this paper, we find all balancing and Lucas-balancing numbers, which can be expressed as the difference of two repdigits. The method of proof involves the application of…

We consider the notion of multiple gap as a finite set of ideals that cannot be separated. We study the different types of such objects that can be found in the Boolean algebra of subsets of the natural numbers modulo finite sets.

Given a rational elliptic surface X over an algebraically closed field, we investigate whether a given natural number k can be the intersection number of two sections of X. If not, we say that k a gap number. We try to answer when gap…

We present a collection of results concerning the location and distribution of very triangular numbers among triangular numbers, including the twin very triangular number theorem, the existence of arbitrarily long gaps between -- and an…

We determine the complete list of the gaps between successive elements of the multiplication table of the first N integers.

The problem of class imbalance is extensive for focusing on numerous applications in the real world. In such a situation, nearly all of the examples are labeled as one class called majority class, while far fewer examples are labeled as the…

We evaluate some types of infinite series with balancing and Lucas-balancing polynomials in closed form. These evaluations will lead to some new curious series for $\pi$ involving Fibonacci and Lucas numbers. Our findings complement those…