No label defined (Q6574105)

The present research deals with \(\varphi\)-expansions of positive integers \(N\) such that are of the following forms: \N\[\NN=\sum^{\infty} _{i=-\infty}{a_i\varphi^i},\N\]\Nwhere \(a_i\in \{0,1\}\) and \(\varphi:= \frac{1+\sqrt 5}{2}\) is the golden mean.\N\N``Even if the number of powers of \( \varphi \) is finite, then any number has infinitely many base phi representations. By not allowing an expansion to end with the digits \(0,1,1\), the number of expansions becomes finite, a solution proposed by Ron Knott. ... We propose another way to obtain finitely many expansions, which we call the natural base phi expansions. We prove that these are closely connected to the Fibonacci partitions''.\N\NFor each natural number \(N\), it is determined a formula for the number of base \(\varphi\)-expansions of the number \(N\) satisfying the equation: \N\[\Na_{R+2}a_{R+1}a_R \ne 011.\N\]\NA behavior of \(\varphi\)-expansions on the odd and the even Lucas interval is investigated, as well as for \(N = F_n\) and \(N = L_n\), simple formulas are given, where \(F_n\) are Fibonacci numbers and \(L_n\) are Lucas numbers. Also, certain additional results, which are related to the Knott, Bergman, and \(\varphi\)-expansions are presented.