The golden ratio is closely related to the Zeckendorf representation which is one of the simplest examples of a numeration system other (other than the usual base-$b$ represenations). As such, it's an important testing ground for new ideas. For instance, in this paper Drmota, Müllner and Spiegelhofer showed that the parity of the sum of digits in the Zeckendorf representation does not correlate with the Möbius function. Such considerations are interesting for their own sake, and are also connected with morphic sequences thanks to the work of Rigo: One can think of a morphic sequence as an automatic sequence in a non-standard numeration system.

The golden ratio is closely related to the Zeckendorf representation which is one of the simplest examples of a numeration system (other than the usual base-$b$ represenations). As such, it's an important testing ground for new ideas. For instance, in this paper Drmota, Müllner and Spiegelhofer showed that the parity of the sum of digits in the Zeckendorf representation does not correlate with the Möbius function. Such considerations are interesting for their own sake, and are also connected with morphic sequences thanks to the work of Rigo: One can think of a morphic sequence as an automatic sequence in a non-standard numeration system.