On tridimensional balancing numbers and some properties

Balancing numbers n and balancers r are originally dened as the solution of the Diophantine equation 1 + 2 + ... + (n - 1) = (n + 1) + (n + 2) + ... + (n + r). If n is a balancing number, then 8n^2 +1 is a perfect square. Further, If n is a balancing number then the positive square root of 8n^2 + 1 is called a Lucas-balancing number. These numbers can be generated by the linear recurrences B_n+1 = 6B_n - B_n-1 and C_n+1 = 6C_n - C_n-1 where B_n and C_n are respectively denoted by the nth balancing number and nth Lucas-balancing number. There is another way to generate balancing and Lucas-balancing numbers using powers of matrices Q_B = (6 -1; 1 0) and Q_C = (17 -3; 3 -1) respectively. The matrix representation, indeed gives many known and new formulas for balancing and Lucas-balancing numbers. In this paper, using matrix algebra we obtain several interesting results on Lucas-balancing numbers.

Balancing numbers n and balancers r are originally defined as the solution of the Diophantine equation 1 + 2 + ··· + (n − 1) = (n + 1) + (n + 2) + ··· + (n + r). If n is a balancing number, then 8n 2 + 1 is a perfect square. Further, If n is a balancing number then the positive square root of 8n 2 + 1 is called a Lucas-balancing number. These numbers can be generated by the linear recurrences Bn+1 = 6Bn − Bn 1 and Cn+1 = 6Cn − Cn 1 where Bn and Cn are respectively denoted by the n th balancing number and n th Lucas-balancing number. In this study, we establish some new identities for the common factors of both balancing and Lucas-balancing numbers. AMS Subject Classification: 11B39, 11B83

\cdots Balancing numbers $n$ and balancers $r$ are originally defined as the solution of the Diophantine equation $1+2+\cdots+(n-1)=(n+1)+(n+2)+\cdots+(n+r)$. These numbers can be generated by the linear recurrence $B_{n+1}=6B_{n}-B_{n-1}$ or by the nonlinear recurrence $B_{n}^{2}=1+B_{n-1} B_{n+1}$. There is another way to generated balancing numbers using powers of a matrix $Q_{B} = \begin{pmatrix} 6 & -1 1 & 0 \end{pmatrix}.$ The matrix representation, indeed gives many known and new formulas for balancing numbers. In this paper, using matrix algebra we obtain several interesting results on balancing and related numbers. Keywords: Balancing numbers; Lucas-balancing numbers; Triangular numbers; Recurrence relation; Balancing Q-matrix; Balancing R-matrix. 2010 Mathematics Subject Classification: 11B39, 11B83

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.

It is well known that if x is a balancing number, then the positive square root of 8x2 + 1 is a Lucas-balancing number. Thus, the totality of balancing number x and Lucas-balancing number y are seen to be the positive integral solutions of the Diophantine equation 8x2 +1 = y2. In this article, we consider some Diophantine equations involving balancing and Lucas-balancing numbers and study their solutions.

In this paper, we find some tridigonal matrices whose determinant and permanent are equal to the negatively subscripted balancing and Lucas- balancing numbers. Also using the First and second kind of Chebyshev polynomials, we obtain the factorization of these numbers.

It is well known that, a recursive relation for the sequence  is an equation that relates  to certain of its preceding terms . Initial conditions for the sequence  are explicitly given values for a finite number of the terms of the sequence. The recurrence relation is useful in certain counting problems like Fibonacci numbers, Lucas numbers, balancing numbers, Lucas-balancing numbers etc. In this study, we use the recurrence relations for both balancing and Lucas-balancing numbers and examine their application to cryptography.

In this work, the general terms of almost balancers, almost cobalancers, almost Lucas-balancers and almost Lucas-cobalancers of first and second type are determined in terms of balancing and Lucas-balancing numbers. Later some relations on all almost balancing numbers and all almost balancers are obtained. Further the general terms of all balancing numbers, Pell numbers and Pell–Lucas number are determined in terms of almost balancers, almost Lucas-balancers, almost cobalancers and almost Lucas-cobalancers of first and second type.

Given a graph $G$, a 2-coloring of the edges of $K_n$ is said to contain a balanced copy of $G$ if we can find a copy of $G$ such that half of its edges is in each color class. If there exists an integer $k$ such that, for $n$ sufficiently large, every 2-coloring of $K_n$ with more than $k$ edges in each color contains a balanced copy of $G$, then we say that $G$ is balanceable. The smallest integer $k$ such that this holds is called the balancing number of $G$.In this paper, we define a more general variant of the balancing number, the generalized balancing number, by considering 2-coverings of the edge set of $K_n$, where every edge $e$ has an associated list $L(e)$ which is a nonempty subset of the color set $\{r,b\}$. In this case, edges $e$ with $L(e) = \{r,b\}$ act as jokers in the sense that their color can be chosen $r$ or $b$ as needed. In contrast to the balancing number, every graph has a generalized balancing number. Moreover, if the balancing number exists, then it coincides with the generalized balancing number.We give the exact value of the generalized balancing number for all cycles except for cycles of length $4k$ for which we give tight bounds. In addition, we give general bounds for the generalized balancing number of non-balanceable graphs based on the extremal number of its subgraphs, and study the generalized balancing number of $K_5$, which turns out to be surprisingly large.

On the properties of k-balancing numbers

To observe the clinical efficacy of auricular acupuncture combined with intermittent catheterization on urethral sphincter overactivity in patients with neurogenic bladder (NB) after spinal cord injury (SCI). A total of 90 patients with SCI were randomly divided into an auricular acupuncture group, an intermittent catheterization group and a combined treatment group, 30 cases in each group. All the patients in the three groups received the basic rehabilitation nursing and normal rehabilitation training. Additionally, in the auricular acupuncture group, auricular acupuncture was applied to "Spleen" (CO13), "Kidney" (CO10), "Bladder" (CO9) and "Subcortex" (AT4). In the intermittent catheterization group, the intermittent catheterization was operated. In the combined treatment group, the auricular acupuncture and the intermittent catheterization were delivered in combination. In the auricular acupuncture and the combined treatment group, the needles were retained for 1 h, once daily. One course of treatment was composed of 6 d and 4 courses were required. At the baseline and after interventions, the maximum urethral pressure (MUP), the residual urine volume (RU), the case number of urethral sphincter opening, the case number of vesicoureteral reflux, the number of white blood cells in urine, the case number of bladder function balance and traditional Chinese medicine (TCM) syndrome score were compared, with clinical efficacy evaluated. Compared with the baseline, MUP, RU, the case number of urethral sphincter opening, the case number of vesicoureteral reflux, the number of white blood cells in urine, the case number of bladder function balance, and TCM symptom score were improved in the combined treatment group after 4 weeks of treatment (P<0.05);and the therapeutic effects of the above indicators except the number of white blood cells in urine and TCM symptom score were improved more significantly when compared with the auricular acupuncture group and the intermittent catheterization group (P<0.05). In the auricular acupuncture group, MUP, RU, the case number of urethral sphincter opening, the case number of vesicoureteral reflux, the case number of bladder function balance and TCM symptom score were ameliorated in comparison with the baseline (P<0.05);while in the intermittent catheterization group, the improvements were obtained in the RU, the case number of urethral sphincter opening, the case number of vesicoureteral reflux, the case number of bladder function balance and the number of white blood cells in urine (P<0.05). The total effective rate in combined treatment group was significantly higher than that in auricular acupuncture group and intermittent catheterization group (P<0.05). The combination of auricular acupuncture and intermittent catheterization is clearly effective on overactivity of the urethral sphincter in the patients with NB after SCI. It can significantly reduce the MUP, the RU, and the incidence of urinary tract infections in the patients, regulate bladder balance status, and attenuate the symptoms of TCM.

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].

In this work, we derive some algebraic identities on generalized Pell numbers and their relationship with balancing numbers. Also we deduce some results on binary quadratic forms involving Pell and balancing numbers.

The balancing numbers and the balancers were introduced by Behera et al. in the year 1999, which were obtained from a simple diophantine equation. The goal of this paper is to investigate the moduli for which all the residues appear with equal frequency with a single period in the sequence of balancing numbers. Also, it is claimed that, the balancing numbers are uniformly distributed modulo 2, and this holds for all other powers of 2 as well. Further, it is shown that the balancing numbers are not uniformly distributed over odd primes.

Let be a graph. A function is said to be a balanced cycle dominating function (BCDF) of if holds for any induced cycle of The balanced cycle domination number of is defined as A graph is said to be a signed cycle balanced graph (SCB-graph) if there exists a function such that holds for any induced cycle of and is said to be a signed cycle balanced dominating function (SCBDF) of The signed cycle balanced domination number of is defined as In this paper, we present upper bounds for balanced cycle domination number and signed cycle balanced domination number. The exact values of this parameter are determined for a few classes of graphs.