On $k$-Fibonacci balancing and $k$-Fibonacci Lucas-balancing numbers [PDF]
Karpatsʹkì Matematičnì Publìkacìï, 2021 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
S.E. Rihane
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Two generalizations of dual-complex Lucas-balancing numbers [PDF]
Acta Universitatis Sapientiae: Mathematica, 2022 In this paper, we study two generalizations of dual-complex Lucas-balancing numbers: dual-complex k-Lucas balancing numbers and dual-complex k-Lucas-balancing numbers.
Bród Dorota +2 more
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On the Reciprocal Sums of Products of Balancing and Lucas-Balancing Numbers [PDF]
Mathematics, 2021 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 ...
Younseok Choo
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Balancing and Lucas-Balancing Numbers and their Application to Cryptography [PDF]
Computer Engineering and Applications Journal, 2016 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.
Sujata Swain +2 more
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Factorizations of negatively subscripted balancing and Lucas-balancing numbers
Boletim da Sociedade Paranaense de Matemática, 2013 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
Prasanta Kumar Ray
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On the Partial Finite Alternating Sums of Reciprocals of Balancing and Lucas-Balancing Numbers
Discussiones Mathematicae - General Algebra and Applications, 2020 In this note, the finite alternating sums of reciprocals of balancing and Lucas-balancing numbers are considered and several identities involving these sums are deduced.
Dutta Utkal Keshari, Ray Prasanta Kumar
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Identities concerning k-balancing and k-Lucas-balancing numbers of arithmetic indexes
AIMS Mathematics, 2019 In this article, we derive some identities involving k balancing and k-Lucas-balancing numbers of arithmetic indexes, say an + p, where a and p are some fixed integers with 0≤p≤a-1.
Prasanta Kumar Ray
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Repdigits as Products of Consecutive Balancing or Lucas-Balancing Numbers [PDF]
The Fibonacci Quarterly, 2018 Repdigits are natural numbers formed by the repetition of a single digit. In this paper, we explore the presence of repdigits in the product of consecutive balancing or Lucas-balancing numbers.
S. G. Rayaguru, G. K. Panda
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Balancing and Lucas-balancing numbers as difference of two repdigits [PDF]
13 pages.
Monalisa Mohapatra +2 more
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TRIGONOMETRIC-TYPE IDENTITIES AND THE PARITY OF BALANCING AND LUCAS-BALANCING NUMBERS [PDF]
TNU Journal of Science and Technology, 2021 Các số cân bằng n được định nghĩa như là nghiệm của phương trình Diophantus 1 + 2 + · · · + (n − 1) = (n + 1) + · · · + (n + r), trong đó r được gọi là hệ số cân bằng ứng với số cân bằng n. Tương tự như vậy, n là một số đối cân bằng với hệ số đối cân bằng r nếu 1 + 2 + · · · + n = (n + 1) + · · · + (n + r).
Ngô Văn Định
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