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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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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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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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$$k$$ 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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