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Panda, G. K.
- On Sum and Ratio Formulas for Balancing Numbers
Abstract Views :216 |
PDF Views:2
Authors
R. K. Davala
1,
G. K. Panda
1
Affiliations
1 Department of Mathematics, National Institute of Technology, Rourkela-769 008, IN
1 Department of Mathematics, National Institute of Technology, Rourkela-769 008, IN
Source
The Journal of the Indian Mathematical Society, Vol 82, No 1-2 (2015), Pagination: 23-32Abstract
This paper deals with the construction of explicit formulas for sum of consecutive balancing numbers, consecutive even/odd balancing numbers, squares of consecutive balancing numbers, squares of consecutive even/odd balancing numbers and pronic product of balancing numbers. Sums of these numbers with alternative signs give beautiful results. When indices are in arithmetic progression, ratios of sums and differences follow certain interesting patterns.Keywords
Balancing and Lucas Balancing Numbers, Balancer, Integer Sequences, Binet Form.
- Almost Balancing Numbers
Abstract Views :238 |
PDF Views:8
Authors
G. K. Panda
1,
A. K. Panda
1
Affiliations
1 Department of Mathematics, National Institute of Technology, Rourkela-769 008, IN
1 Department of Mathematics, National Institute of Technology, Rourkela-769 008, IN
Source
The Journal of the Indian Mathematical Society, Vol 82, No 3-4 (2015), Pagination: 147-156Abstract
Almost balancing numbers are defined from a Diophantine equation slightly different from the defining equation for balancing numbers. There are two types of almost balancing numbers and are respectively the ceiling and floor functions of square ischolar_mains of two types of almost square triangular numbers. These numbers are very closely associated with balancing, Lucas balancing, Pell and associated Pell numbers.Keywords
Triangular Numbers, Balancing Numbers, Cobalancing Numbers, Pell Numbers, Associate Pell Numbers.- Sum Formulas Involving Powers of Balancing and Lucas-balancing Numbers
Abstract Views :257 |
PDF Views:0
Authors
Affiliations
1 Department of Mathematics, National Institute of Technology, Rourkela, IN
1 Department of Mathematics, National Institute of Technology, Rourkela, IN
Source
The Journal of the Indian Mathematical Society, Vol 86, No 1-2 (2019), Pagination: 137-160Abstract
In this article, we obtain the closed form expressions for different types of summation formulas involving certain powers of balancing and Lucas-balancing numbers using the telescoping summation formula.Keywords
Balancing Numbers, Lucas-balancing Numbers, Telescoping Summation Formula.References
- Adegoke, K., Factored closed-form expressions for the sums of cubes of Fibonacci and Lucas numbers, https://arxiv.org/pdf/1706.08469.pdf.
- Adegoke, K., Sums of fourth powers of Fibonacci and Lucas numbers, https://arxiv.org/pdf/1706.00407.pdf.
- Behera, A., Panda, G.K., On the square ischolar_mains of triangular numbers, Fib. Quart., 37(2)(1999), 98–105.
- Clary, S., Hemenway, P.D., On sums of cubes of Fibonacci numbers, in Applications of Fibonacci Numbers, Kluwer Academic Publishers, Dordrecht, The Netherlands, 5 (1993), 123–136.
- Davala, R.K., Panda, G.K., On Sum and Ratio Formulas for Balancing Numbers, Journal of the Ind. Math. Soc., 82(1–2) (2015).
- Melham, R.S., Alternating sums of fourth powers of Fibonacci and Lucas numbers, Fib. Quart., 38(3) (2000), 254–259.
- Panda, G.K., Some fascinating properties of balancing numbers, In Proc. of Eleventh Internat. Conference on Fibonacci Numbers and Their Applications, Cong. Numerantium, 194 (2009), 185–189.
- Shift Balancing Numbers
Abstract Views :261 |
PDF Views:1
Authors
Affiliations
1 Department of Mathematics, National Institute of Technology, Rourkela - 769 008, IN
2 Department of Mathematics, Gayatri Vidya Parishad College of Engineering (A), Visakhapatnam - 530048, IN
1 Department of Mathematics, National Institute of Technology, Rourkela - 769 008, IN
2 Department of Mathematics, Gayatri Vidya Parishad College of Engineering (A), Visakhapatnam - 530048, IN
Source
The Journal of the Indian Mathematical Society, Vol 87, No 1-2 (2020), Pagination: 131–136Abstract
For each positive integer k, the Diophantine equation (k+1)+(k+2)+···+(n−1) = (n+1)+(n+2)+···+(n+r) is studiedKeywords
Lancing Numbers, Lucas-Balancing Numbers, Gap Balancing Numbers, t−balancing Numbers.References
- A. Behera and G. K. Panda, On the square ischolar_mains of triangular numbers, Fibonacci Quart., 37(2) (1999), 98-105.
- K. K. Dash and R. S. Ota, t−Balancing Numbers, Int. J. Contemp. Math. Sciences, 7 (2012), 1999- 2012.
- R. K. Davala and G. K. Panda, On sum and ratio formulas for balancing numbers, J. Ind. Math. Soc., 82(1-2) (2015), 23-32.
- R. P. Finkelstein, The House Problem, Amer. Math. Monthly, 72 (1965), 1082-1088.
- T. Kovács, K. Liptai and P. Olajos, On (a,b)-balancing numbers, Publ. Math. Debrecen, 77(3) (2010).
- K. Liptai, Fibonacci Balancing Numbers, Fibonacci Quart., 42(4) (2004), 330-340.
- K. Liptai, Lucas balancing numbers, Commun. Math., 14(1) (2006), 43-47.
- R. A. Mollin, Fundamental number theory with applications, Boca Raton, CRC press, London (2004).
- G. K. Panda and R. K. Davala, Perfect balancing numbers, Fibonacci Quart., 53(3) (2015), 261-264.
- G. K. Panda and P. K. Ray, Cobalancing numbers and cobalancers, Internat. J. Math. Math. Sci., 8 (2005), 1189-1200.
- G. K. Panda, Some fascinating properties of balancing numbers, In Proc. of Eleventh Internat. Conference on Fibonacci Numbers and Their Applications, Cong. Numerantium, 194, (2009), 185-189.
- G. K. Panda and A. K. Panda, Almost balancing numbers, J. Ind. Math. Soc., 82(3-4) (2015), 147-156.
- G. K. Panda and S. S. Rout, Periodicity of balancing numbers, Acta. Math. Hungar., 143(2) (2014), 274-286.
- S. S. Rout and G. K. Panda, k-gap balancing numbers, Mathematika, 70(1) (2015), 109-121.
- S. S. Rout, R. K. Davala and G. K. Panda, Stability of balancing sequence modulo p, Unif. Distrib. Theory, 10(2) (2015), 77-91.
- Brousseau’s Reciprocal Sums Involving Balancing and Lucas-Balancing Numbers
Abstract Views :103 |
PDF Views:0
Authors
Affiliations
1 Department of Center for Data Science, Siksha ’O’ Anusandhan Deemed to be University, Bhubaneswar - 751 030, IN
2 Department of Mathematics, National Institute of Technology, Rourkela - 769 008, IN
1 Department of Center for Data Science, Siksha ’O’ Anusandhan Deemed to be University, Bhubaneswar - 751 030, IN
2 Department of Mathematics, National Institute of Technology, Rourkela - 769 008, IN
Source
The Journal of the Indian Mathematical Society, Vol 89, No 1-2 (2022), Pagination: 145–166Abstract
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.
Keywords
Balancing numbers, Lucas-balancing numbers, gap balancing numbers, t-balancing numbers.References
- K. Adegoke, Generalizations of the Reciprocal Fibonacci-Lucas Sums of Brousseau, J. Integer Seq., Vol. 21 (2018), Article 18.1.6.
- A. Behera and G. K. Panda, On the square roots of triangular numbers, Fibonacci Quart., 37(2) (1999), 98–105.
- Bro. A. Brousseau, Summation of Infinite Fibonacci series, Fib. Quart., 7 (1969), 143–168.
- Bro. A. Brousseau, Fibonacci-Lucas Infinite Series-Research Topic, Fib. Quart., 7 (1969), 211–217.
- R. K. Davala and G. K. Panda, On Sum and Ratio Formulas for Balancing Numbers, J. Indian Math. Soc., 82(1-2) (2015), 23–32.
- R. Frontczak, New results on reciprocal series related to Fibonacci and Lucas numbers with subscripts in arithmetic progression, Int. J. Contemporary Math. Sci., 11 (2016), 509–516.
- G. K. Panda, Some fascinating properties of balancing numbers, In Proc. of Eleventh Internat. Conference on Fibonacci Numbers and Their Applications, Cong. Numerantium, 194 (2009), 185–189.
- S. G. Rayaguru and G. K. Panda, Some infinite product identities involving balancing and Lucas-balancing numbers, Alabama J. Math., 42 (2018).
- S. G. Rayaguru and G. K. Panda, Sum Formulas Involving Powers of Balancing and Lucas-balancing Numbers, J. Indian Math. Society, 86(1-2) (2019), 1–24.
- S. G. Rayaguru and G. K. Panda, Sum Formulas Involving Powers of Balancing and Lucas-balancing Numbers-II, Notes on Number Theory and Discrete Mathematics, 25(3) (2019), 102–110.