On the Properties of Lucas-Balancing Numbers by Matrix Method

Authors

  • Prasanta Kumar Ray IIIT BhubaneswarIndia

Keywords:

Balancing numbers, Lucas-balancing numbers, Balancing matrix, Lucas-balancing matrix

Abstract

Balancing numbers n and balancers r are originally dened as the solution of the Diophantine equation 1 + 2 + ... + (n - 1) = (n + 1) + (n + 2) + ... + (n + r). If n is a balancing number, then 8n^2 +1 is a perfect square. Further, If n is a balancing number then the positive square root of 8n^2 + 1 is called a Lucas-balancing number. These numbers can be generated by the linear recurrences B_n+1 = 6B_n - B_n-1 and C_n+1 = 6C_n - C_n-1 where B_n and C_n are respectively denoted by the nth balancing number and nth Lucas-balancing number. There is another way to generate balancing and Lucas-balancing numbers using powers of matrices Q_B = (6 -1; 1 0) and Q_C = (17 -3; 3 -1) respectively. The matrix representation, indeed gives many known and new formulas for balancing and Lucas-balancing numbers. In this paper, using matrix algebra we obtain several interesting results on Lucas-balancing numbers.

Author Biography

Prasanta Kumar Ray, IIIT BhubaneswarIndia

Mathematics

Assistant Professor

References

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Published

09-02-2014

How to Cite

Ray, P. K. (2014). On the Properties of Lucas-Balancing Numbers by Matrix Method. Sigmae, 3(1), 1–6. Retrieved from https://publicacoes.unifal-mg.edu.br/revistas/index.php/sigmae/article/view/149

Issue

Section

Pure Mathematics