The structure of the expansions
Abstract
The expansions of f in the unit of six Rf‘s have a distinct structure. Therefore a alternative notation for the expansions will be introduced: only the number of 0’s will be mentioned. The even and odd part of the Fibonacci related sequence f will have its own expansion. The basic form of these expansions is based on ba-cd.
The odd part of the expansions of the second Rf of the unit of six Rf‘s contains the values of ba-cd. The development of the sequence of ba-cd‘s has a structure based on Fibonacci intervals. Also the development of the sequences a-a’ and b-b’ will be discussed.
A same kind of structure can be recognized in the structure of the optimal (canonical) expansions.
Special to the a and c of ba-cd is that they develope as a nested Fibonacci word.
The structure of the value of Rf is Fibonacci based.
The expansions of Rf in the unit of six Rf‘s have the same structure as long as they are based on the Fibonacci sequence. A part of Rf equals expansions of f.
Finally there seems to be a similar structure for the natural numbers.
- 1 The expansions of f
- 2 The sequence of ba-cd's
- 3 The sequences a-a' and b-b'
- 4 A same kind of structure
- 5 A nested Fibonacci word
- 6 The values of R(f)
- 7 The expansions of R(f)
- 8 The natural numbers
