Aitken sequences and generalized Fibonacci numbers

Abstract:

Consider the sequence $({v_n})$ generated by ${v_{n + 1}} = a{v_n} - b{v_{n - 1}}$, $n \geqslant 2$, where ${v_1} = 1$, ${v_2} = a$, with a and b real, of which the Fibonacci sequence is a special case. It is shown that if Aitken acceleration is used on the sequence $({x_n})$ defined by ${x_n} = {v_{n + 1}}/{v_n}$, the resulting sequence is a subsequence of $({x_n})$. Second, if Newton’s method and the secant method are used (with suitable starting values) to solve the equation ${x^2} - ax + b = 0$, then the sequences obtained from both of those methods are also subsequences of the original sequence.