$$k$$ k -Gap balancing numbers

Rout, Sudhansu Sekhar; Panda, G. K.

doi:10.1007/s10998-014-0067-7

Cited by 10 publications

(10 citation statements)

References 4 publications

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“…However, for many values of D, D-subbalancer numbers exist. For example, one can verify that a natural number x is a 6−subbalancing number if and only if 8x 2 + 49 is a perfect square and the values of x satisfying 8x 2 + 49 = y 2 are 5−gap balancing numbers [4]. Indeed, finding all feasible values of D is an interesting problem.…”

Section: Discussionmentioning

confidence: 99%

“…If k = 2 then b k = 2 and the requirement for a positive integer n to be a b 2 -subbalancing number is that 8n 2 + 17 be a perfect square. But according to Rout and Panda [4], such numbers are 3-gap balancing numbers and are of the form 5B n ± C n .…”

Section: Subbalancing Numbersmentioning

confidence: 99%

“…Rout and Panda [4] generalized the concept of balancing numbers and introduced gap balancing numbers. If k is odd, they call a natural number n a k-gap balancing number if…”

Section: Introductionmentioning

confidence: 99%

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Subbalancing Numbers

Davala

Panda

2018

MATEMATIKA

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A natural number $n$ is called balancing number (with balancer $r$)if it satisfies the Diophantine equation $1+2+\cdots+(n-1)=(n+1)+(n+2)+\cdots+(n+r).$ However, if for some pair of natural numbers $(n,r)$, $1+2+\cdots+(n-1) < (n+1)+(n+2)+\cdots+(n+r)$ and equality is achieved after adding a natural number $D$ to the left hand side then we call $n$ a $D$-subbalancing number with $D$-subbalaner number $r$. In this paper, such numbers are studied for certain values of $D$.

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Section: Discussionmentioning

confidence: 99%

Section: Subbalancing Numbersmentioning

confidence: 99%

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Subbalancing Numbers

Davala

Panda

2018

MATEMATIKA

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“…Rout and Panda [18] generalized the concept of balancing numbers and introduced gap balancing numbers. If k is odd, they call a natural number n a k-gap balancing number if…”

Section: Introductionmentioning

confidence: 99%

A generalization to almost balancing and cobalancing numbers using triangular numbers

Rayaguru¹,

Panda²

2020

NNTDM

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A generalization of almost balancing numbers is studied using triangular numbers as the difference between the left and right hand sides of the defining equation of balancing numbers. In case of almost balancing numbers, this difference is kept 1, which is the first triangular number. Some specific representations of these numbers in terms of balancing and balancing related numbers are established and few more results with triangular, square triangular, balancing and balancing related numbers are also studied so as to generalize the identities obtained by A. Tekcan.

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“…is the ith triangular number. Interest in balancing numbers [1] and their generalizations [2,3,5,6,7,10,11,12,13,14,15] stems from contemporary investigations into the properties of square triangular numbers and related expressions. A central theme is studying a geometrically motivated sequences through solutions to associated Pell-like equations.…”

Section: Introductionmentioning

confidence: 99%

Classes of Gap Balancing Numbers

Bartz¹,

Dearden²,

Iiams³

2018

Preprint

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Gap balancing numbers are a certain generalization of balancing and cobalancing numbers that arise from studying the equation T (L) + T (B) = T (m) where T (i) is the ith triangular number. In this paper, we survey early results, attempt to unify existing terminology, and extend prior findings. In addition, we further investigate the structure of classes of gap balancing numbers and related sequences, presenting new results and a conjecture regarding the number of classes based on the gap size. Definition 2.1. Let k ≥ 0 be an integer. Define a positive integer B to be an upper k-gap balancing number if 1 + 2 + 3 + • • • + (B − k) = (B + 1) + • • • + (B + r)

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$$k$$ k -Gap balancing numbers

Cited by 10 publications

References 4 publications

Subbalancing Numbers

Subbalancing Numbers

A generalization to almost balancing and cobalancing numbers using triangular numbers

Classes of Gap Balancing Numbers

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