On the Properties of Lucas-Balancing Numbers by Matrix Method

  • Prasanta Kumar Ray IIIT Bhubaneswar India


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


Assistant Professor


Behera, A. and Panda, G.K. On the square roots of triangular numbers, The Fibonacci Quarterly,

(2), 1999, 98 - 105.

Liptai, K. Fibonacci balancing numbers, The Fibonacci Quarterly, 42(4), 2004, 330-340.

Panda, G. K. and Ray, P. K. Some links of balancing and cobalancing numbers with Pell and

associated Pell numbers, Bulletin of the Institute of Mathematics, Academia Sinica (New Series),

(1), 2011, 41-72.

Panda, G. K. Some fascinating properties of balancing numbers, in Proc. Eleventh Internat.

Conference on Fibonacci Numbers and Their Applications, Cong. Numerantium, 194, 2009: 185-

Ray, P. K. Application of Chybeshev polynomials in factorization of balancing and Lucas-balancing

Numbers, Bol. Soc. Paran. Mat. Vol.30 (2), 2012, 49-56.

Ray, P. K. Factorization of negatively subscripted balancing and Lucas-balancing numbers,Bol.

Soc. Paran. Mat., Vol.31 (2), 2013, 161-173.

Ray, P. K. Certain matrices associated with balancing and Lucas-balancing numbers, Matematika,

Vol.28 (1), 2012, 15-22