A trigonometry approach to balancing numbers and their related sequences

The balancing numbers satisfy the second order linear homogeneous difference equation Bn+1 = 6B(n) - Bn-1 on the other hand the Fibonacci numbers are solution of the second order linear homogeneous difference equation Fn+1 = F-n + Fn-1, where Bn and F-n denote the nth balancing number and nth Fibonacci number respectively. In a paper, Smith introduced Fibonometry in connection with a differential equation called Fibonometric differential equation. In this study, we first introduce the balancometric differential equation and then obtain the balancometric functions as solutions of this equation.