On the Properties of Lucas-Balancing Numbers by Matrix Method

  • Prasanta Kumar Ray IIIT Bhubaneswar India

Resumo

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.

Biografia do Autor

Prasanta Kumar Ray, IIIT Bhubaneswar India

Mathematics

Assistant Professor

Referências

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Publicado
09-02-2014
Seção
Mathematics