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Balancing Numbers as Sum of Same Power of Consecutive Balancing Numbers
Vietnam Journal of Mathematics, 2022The sequence of balancing numbers \( \{B_n\}_{n\ge 0} \) is defined by the binary recurrence \( B_0=0 \), \( B_1=1 \), and \( B_{n+!}=6B_n-B_{n-1} \) for all \( n\ge 1 \). In the paper under review, the authors study the Diophantine equation \[ B_n^{x}+B_{n+1}^{x}+\cdots +B_{n+k-1}^{x}=B_m,\tag{1} \] in positive integers \( (m,n,k,x) \).
Souleymane Nansoko +2 more
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Periodicity of Balancing Numbers
Acta Mathematica Hungarica, 2014The balancing numbers originally introduced by \textit{A. Behera} and \textit{G. K. Panda} [Fibonacci Q. 37, No. 2, 98--105 (1999; Zbl 0962.11014)] as solutions of a Diophantine equation on triangular numbers possess many interesting properties. Many of these properties are comparable to certain properties of Fibonacci numbers, while some others are ...
Panda, G. K., Rout, S. S.
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Publicationes Mathematicae Debrecen, 2010
A positive integer \(n\) is a balancing number if \(1 +\dots + (n - 1) = (n + 1) + \dots + (n + r)\) holds with some positive integer \(r\). The problem of finding balancing numbers goes back to the work of \textit{R. Finkelstein} [Am.\ Math.\ Mon.\ 72, 1082--1088 (1965; Zbl 0151.03305)].
Kovács, Tünde +2 more
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A positive integer \(n\) is a balancing number if \(1 +\dots + (n - 1) = (n + 1) + \dots + (n + r)\) holds with some positive integer \(r\). The problem of finding balancing numbers goes back to the work of \textit{R. Finkelstein} [Am.\ Math.\ Mon.\ 72, 1082--1088 (1965; Zbl 0151.03305)].
Kovács, Tünde +2 more
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$$k$$ k -Gap balancing numbers
Periodica Mathematica Hungarica, 2015A natural number \(n\) is called a balancing number with balancer \(r\) if \[ 1 + 2 + \ldots + (n - 1) = (n + 1) + (n + 2) + \ldots + (n + r). \] On other hand \(n\) is called a cobalancing number with cobalancer \(r\) if \[ 1 + 2 + \ldots + n = (n + 1) + (n + 2) + \ldots + (n + r). \] Several papers in this area are presently available.
Sudhansu Sekhar Rout +1 more
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On the number of balanced signed graphs
The Bulletin of Mathematical Biophysics, 1967The classical enumeration theorem of Polya (Acta Math.,68, 145–254, 1937) is applied to a modified version of Harary’s (Pacific J. Math.,8, 743–755, 1958) generating functions for counting bicolored graphs to derive a counting function for the number of balanced signed graphs. Methods for computing these counting polynomial functions are discussed.
Harary, Frank, Palmer, Edgar M.
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Factoriangular numbers in balancing and Lucas-balancing sequence
Boletín de la Sociedad Matemática Mexicana, 2020The balancing numbers \(\{B_n\}_{n\ge 0}\) have initial terms \(B_0=0,~B_1=1\) and satisfy the recurrence \(B_{n+2}=6B_{n+1}-B_n\) for all \(n\ge 0\). The Lucas-balancing numbers \(\{C_n\}_{n\ge 0}\) have initial terms \(C_0=1,~C_1=3\) and satisfy the same recurrence relation as the balancing numbers.
Sai Gopal Rayaguru +2 more
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The maximum number of balancing sets
Graphs and Combinatorics, 1987Let \(a_ 1,...,a_ n\) be a sequence of nonzero real numbers such that \(\sum^{n}_{i=1}a_ i=0\). B is called a balancing set if \(\sum_{b\in B}a_ b=0\). Let f(n) be the maximum number of balancing sets. It is shown that \(f(n)=\left( \begin{matrix} 2k\\ k\end{matrix} \right)\) if \(n=2k\) and \(f(n)=2\left( \begin{matrix} 2k\\ k-1\end{matrix} \right ...
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On the Periodicity of Lucas-Balancing Numbers and p-adic Order of Balancing Numbers
Iranian Journal of Science and Technology, Transactions A: Science, 2020The objective of this article is to study the periodicity of Lucas-balancing numbers modulo any positive integer. Some relations between the periodicity of balancing and Lucas-balancing numbers are also discussed. Further, in this study the p-adic order of balancing numbers is completely characterized .
Takao Komatsu +2 more
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INCOMPLETE BALANCING AND LUCAS-BALANCING NUMBERS
2018The aim of this article is to establish some combinatorial expressions of balancing and Lucas-balancing numbers and investigate some of their properties.
Patel, Bijan Kumar +2 more
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Ars Combinatoria
Behera and Panda defined a balancing number as a number b for which the sum of the numbers from 1 to b – 1 is equal to the sum of the numbers from b + 1 to b + r for some r. They also classified all such numbers. We define two notions of balancing numbers for Farey fractions and enumerate all possible solutions.
Noah Lebowitz-Lockard, Joseph Vandehey
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Behera and Panda defined a balancing number as a number b for which the sum of the numbers from 1 to b – 1 is equal to the sum of the numbers from b + 1 to b + r for some r. They also classified all such numbers. We define two notions of balancing numbers for Farey fractions and enumerate all possible solutions.
Noah Lebowitz-Lockard, Joseph Vandehey
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