On the frequency of occurrence of \(\alpha^i\) in the \(\alpha\)-expansions of the positive integers (Q2719873)

Let \(\alpha=\frac{1+\sqrt{5}}{2}\) be the golden ratio. Any positive integer \(n\) can be represented ``in the base \(\alpha\)'' in the form \(n=\sum_{i=-\ell(n)}^{u(n)} e_i(n)\alpha^i\), where \(\ell(n), u(n)\) are positive integers and \(e_i(n)\in \{0,1\}\) [see \textit{P. J. Grabner, R. F. Tichy, I. Nemes} and \textit{A. Pethő}, Appl. Math. Lett. 7, 25-28 (1994; Zbl 0792.11002)]. NEWLINENEWLINENEWLINEThe authors of the present paper investigate the density of the subset of positive integers \(n\) such that \(e_i(n)=1\) for a fixed \(i\). They deduce a general formula for this density using combinatorial and algorithmic techniques and properties of the Fibonacci and Lucas numbers.