How to sum powers of balancing numbers efficiently

Helmut Prodinger
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 27, 2021, Number 1, Pages 134—137
DOI: 10.7546/nntdm.2021.27.1.134-137
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Authors and affiliations

Helmut Prodinger
Department of Mathematical Sciences, Stellenbosch University
7602 Stellenbosch, South Africa

Abstract

Balancing numbers possess, as Fibonacci numbers, a Binet formula. Using this, partial sums of arbitrary powers of balancing numbers can be summed explicitly. For this, as a first step, a power B_n^l is expressed as a linear combination of B_{mn}.

Keywords

  • Balancing numbers
  • Binet formula
  • Generating functions

2010 Mathematics Subject Classification

  • 11B49
  • 05A15

References

  1. The online encyclopedia of integer sequences. http://oeis.org.
  2. Prodinger, H. (2020). Sums of powers over equally spaced Fibonacci numbers. Integers, paper A37.
  3. Prodinger, H. (2021). Summing a family of generalized Pell numbers. Annales Mathematicae Silesianae, doi: https://doi.org/10.2478/amsil-2020-0024.
  4. Komatsu, T. (2020). Higher-order identities for balancing numbers. Notes on Number Theory and Discrete Mathematics, 26(2), 71–84

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Cite this paper

Prodinger, H. (2021). How to sum powers of balancing numbers efficiently. Notes on Number Theory and Discrete Mathematics, 27(1), 134-137, doi: 10.7546/nntdm.2021.27.1.134-137.

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