(untitled) (MATEMATIKA, 2012)
… Balancing numbers n and balancers r are originally defined as the solution of the Diophantine equation 1+2+…+(n-1)=(n+1)+(n+2)+…+(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 = pmatrix 6 & -1 1 & 0 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