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Is there an irrational number containing only $0$'s and $1$'s with continued fraction entries less than $10$?

I asked for an irrational number with digits $0$ and $1$, such that the continued fraction has no entry larger than $9$

The number

$$0.1011100100100000000000100000\cdots$$

with ones at the positions

1 3 4 5 8 11 23 38 77 155 311 623 1247 2495 ...

seems to satisfy the condition. From $38$ on, the next position is at $2p+1$, if the previous position is at $p$

The finite decimal expansion (truncation at the $1$ in position $2495$) has the property that no entry of the continued fraction is larger than $9$, but what about the irrational number formed this way ? Can we prove that it has the property as well ?

  • 0
    If we truncate the number at any $1$ at position $4$ to $638975$, we get a number with the following properties : $1)$ As already mentioned, the continued fraction contains no entry larger than $9$ , $2)$ The continued fraction contains no $7$ – 2017-01-20

0 Answers 0