Balances in the Set of Arithmetic Progressions

This article focuses on searching and classifying balancing numbers in a set of arithmetic progressions. The sufficient and necessary conditions for the existence of balancing numbers are presented. Moreover, explicit formulae of balancing numbers and various relations are included.

TRIGONOMETRIC-TYPE IDENTITIES AND THE PARITY OF BALANCING AND LUCAS-BALANCING NUMBERS

Các số cân bằng n được định nghĩa như là nghiệm của phương trình Diophantus 1 + 2 + · · · + (n − 1) = (n + 1) + · · · + (n + r), trong đó r được gọi là hệ số cân bằng ứng với số cân bằng n. Tương tự như vậy, n là một số đối cân bằng với hệ số đối cân bằng r nếu 1 + 2 + · · · + n = (n + 1) + · · · + (n + r). Ký hiệu Bn là số cân bằng thứ n và bn là số đối cân bằng thứ n. Khi đó, 8Bn2 + 1 và 8b 2n +8bn +1 là những số chính phương. Số Lucas-cân bằng thứ n, ký hiệu Cn, và số Lucas-đối cân bằng thứ n, ký hiệu cn, lần lượt là các căn bậc hai dương của 8Bn2 + 1 và 8b 2n + 8bn + 1. Trong bài báo này, bằng những tính toán sơ cấp, chúng tôi thiết lập một số đẳng thức kiểu lượng giác và từ đó chỉ ra một số tính chất số học liên quan đến tính chẵn lẻ của các số cân bằng, các số đối cân bằng, các số Lucas-cân bằng và các số Lucas-đối cân bằng.

Markoff–Rosenberger triples and generalized Lucas sequences

We consider the Markoff–Rosenberger equation $$\begin{aligned} ax^2+by^2+cz^2=dxyz \end{aligned}$$ a x 2 + b y 2 + c z 2 = d x y z with $$(x,y,z)=(U_i,U_j,U_k)$$ ( x , y , z ) = ( U i , U j , U k ) , where $$U_i$$ U i denotes the i-th generalized Lucas number of first/second kind. We provide an upper bound for the minimum of the indices and we apply the result to completely resolve concrete equations, e.g. we determine solutions containing only balancing numbers and Jacobsthal numbers, respectively.

A Study on Generalized Balancing Numbers

In this paper, we investigate properties of the generalized balancing sequence and we deal with, in detail, namely, balancing, modified Lucas-balancing and Lucas-balancing sequences. We present Binet’s formulas, generating functions and Simson formulas for these sequences. We also present sum formulas of these sequences. We provide the proofs to indicate how the sum formulas, in general, were discovered. Of course, all the listed sum formulas may be proved by induction, but that method of proof gives no clue about their discovery. Moreover, we consider generalized balancing sequence at negative indices and construct the relationship between the sequence and itself at positive indices. This illustrates the recurrence property of the sequence at the negative index. Meanwhile, this connection holds for all integers. Furthermore, we give some identities and matrices related with these sequences.

On $k$-Fibonacci balancing and $k$-Fibonacci Lucas-balancing numbers

The balancing number $n$ and the balancer $r$ are solution of the Diophantine equation $$1+2+\cdots+(n-1) = (n+1)+(n+2)+\cdots+(n+r). $$ It is well known that if $n$ is balancing number, then $8n^2 + 1$ is a perfect square and its positive square root is called a Lucas-balancing number. For an integer $k\geq 2$, let $(F_n^{(k)})_n$ be the $k$-generalized Fibonacci sequence which starts with $0,\ldots,0,1,1$ ($k$ terms) and each term afterwards is the sum of the $k$ preceding terms. The purpose of this paper is to show that 1, 6930 are the only balancing numbers and 1, 3 are the only Lucas-balancing numbers which are a term of $k$-generalized Fibonacci sequence. This generalizes the result from [Fibonacci Quart. 2004, 42 (4), 330-340].

On Balancing and Lucas-balancing Quaternions

The aim of this article is to investigate two new classes of quaternions, namely, balancing and Lucas-balancing quaternions that are based on balancing and Lucas-balancing numbers, respectively. Further, some identities including Binet’s formulas, summation formulas, Catalan’s identity, etc. concerning these quaternions are also established.

How to sum powers of balancing numbers efficiently

Balancing numbers possess, as Fibonacci numbers, a Binet formula. Using this, partial sums of arbitrary powers of balancing numbers can be summed explicitly. For this, as a first step, a power B_n^l is expressed as a linear combination of B_{mn}.

On the Reciprocal Sums of Products of Balancing and Lucas-Balancing Numbers

Recently Panda et al. obtained some identities for the reciprocal sums of balancing and Lucas-balancing numbers. In this paper, we derive general identities related to reciprocal sums of products of two balancing numbers, products of two Lucas-balancing numbers and products of balancing and Lucas-balancing numbers. The method of this paper can also be applied to even-indexed and odd-indexed Fibonacci, Lucas, Pell and Pell–Lucas numbers.