Certain Matrices Associated with Balancing and Lucas-balancing Numbers

Abstract

\cdots Balancing numbers $n$ and balancers $r$ are originally defined as the solution of the Diophantine equation $1+2+\cdots+(n-1)=(n+1)+(n+2)+\cdots+(n+r)$. These numbers can be generated by the linear recurrence $B_{n+1}=6B_{n}-B_{n-1}$ or by the nonlinear recurrence $B_{n}^{2}=1+B_{n-1} B_{n+1}$. There is another way to generated balancing numbers using powers of a matrix $Q_{B} = \begin{pmatrix} 6 & -1 \\ 1 & 0\\ \end{pmatrix}.$ The matrix representation, indeed gives many known and new formulas for balancing numbers. In this paper, using matrix algebra we obtain several interesting results on balancing and related numbers. Keywords: Balancing numbers; Lucas-balancing numbers; Triangular numbers; Recurrence relation; Balancing Q-matrix; Balancing R-matrix. 2010 Mathematics Subject Classification: 11B39, 11B83