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Golden ratio base

Let $\phi$ be the golden ratio and look at real numbers as expansions in digits from base $\phi + 1$. Has this base been considered or studied anywhere? … The rule for when a digit is too large is also simple (a sum of a sequence of $n$ digits may not exceed $n+1$). So we count $1$, $2$, $10.1$, $11.1$, $12.1$, $20.2$, $100.01$ etc. …
Maarten Havinga's user avatar