On Some Identities of Balancing-Like and Lucas-Balancing-Like Sequences by the Matrix Method

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Published: Nov 1, 2023
Keywords:
balancing-like sequence Lucas-balancing-like sequence matrix method

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Sanit Ritthidej
Faculty of Science and Technology, Songkhla Rajabhat University, Songkhla
Theerapon Buatong
Faculty of Science and Technology, Songkhla Rajabhat University, Songkhla
Sirachat Tipsri
Faculty of Science and Technology, Songkhla Rajabhat University, Songkhla

Abstract

Balancing-like and Lucas-balancing-like sequences are defined by gif.latex?\left&space;\{&space;\frac{\sigma&space;_k}{2}&space;\right&space;\}  and gif.latex?\left&space;\{&space;N_k&space;\right&space;\}  for positive integer gif.latex?k , respectively,  for gif.latex?N_k and gif.latex?\sigma&space;_k  are solutions of Pell’s equation gif.latex?N^2-12\sigma^2=1. These sequences are determined as the linear recurrence relations gif.latex?L_{n+1}=4L_{n}-L_{n-1}  and gif.latex?M_{n+1}=4M_{n}-M_{n-1}  with initial conditions gif.latex?L_{0}=0,L_{1}=1 and gif.latex?M_{0}=1,M_{1}=2  for positive integer gif.latex?n  , respectively. In this paper, some identities of Balancing-like and Lucas-balancing-like sequences are obtained by using Binet’s formulas which are formulated by matrices gif.latex?Q_{L} and gif.latex?Q_{M}.

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How to Cite
Ritthidej, S., Buatong, T., & Tipsri, S. (2023). On Some Identities of Balancing-Like and Lucas-Balancing-Like Sequences by the Matrix Method. PKRU SciTech Journal, 7(2), 37–48. Retrieved from https://ph01.tci-thaijo.org/index.php/pkruscitech/article/view/252191
Issue
Vol. 7 No. 2 (2023): July - December
Section
Research Articles
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References

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Panda, G. K. (2009). Some fascinating properties of balancing numbers (pp.185-189). In Proceedings of the Eleventh International Conference on Fibonacci Numbers and Their Applications. Cong. Numerantium.

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Panda, G. K., & Panda, A. K. (2014). Balancing-like sequences associated with integral standard deviations of consecutive natural numbers. Applied Mathematical Sciences, 52(5), 187-192.

Minc, H. (1978). Permanents Encyclopaedia of Mathematics and its Applications, vol 6. London: Addison-Wesley Publishing Company.

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