Results 281 to 290 of about 1,156,336 (332)

Balancing Numbers as Sum of Same Power of Consecutive Balancing Numbers

Vietnam Journal of Mathematics, 2022
The 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, Euloge Tchammou, Alain Togbé   +2 more
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Polygonal Balancing Numbers I

2022
See the abstract in the attached pdf.
Bartz, Jeremiah, Dearden, Bruce, Iiams, Joel   +2 more
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Brousseau’s Reciprocal Sums Involving Balancing and Lucas-Balancing Numbers

The Journal of the Indian Mathematical Society, 2022
In this paper, we derive the closed form expressions for the finite and infinite sums with summands having products of balancing and Lucas-balancing numbers in the denominator. We present some generalized Brousseau’s sums for balancing and Lucas-balancing numbers.
Rayaguru, S. G., Panda, G. K.
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Periodicity of Balancing Numbers

Acta Mathematica Hungarica, 2014
The 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, 2015
A 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.
Rout, S. S., Panda, G. K.
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Factoriangular numbers in balancing and Lucas-balancing sequence

Boletín de la Sociedad Matemática Mexicana, 2020
The 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, Japhet Odjoumani, Gopal Krishna Panda   +2 more
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On $(a,b)$-balancing numbers

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, Liptai, Kálmán, Olajos, Péter   +2 more
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Diophantine equations concerning balancing and Lucas balancing numbers

Archiv der Mathematik, 2016
zbMATH Open Web Interface contents unavailable due to conflicting licenses.
Dey, Pallab Kanti, Rout, Sudhansu Sekhar
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