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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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Brousseau’s Reciprocal Sums Involving Balancing and Lucas-Balancing Numbers
The Journal of the Indian Mathematical Society, 2022In 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, 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.
Rout, S. S., Panda, G. K.
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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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