Representation of powers of three in binary. Do powers of $3$ have a simple representation in binary ? I fail to spot an obvious pattern or anything else. The only thing I can think of is using the binomial theorem
$$3^n = (1+2)^n = \sum_{k=0}^n {{n}\choose{k}}2^k 1^{n-k} = \sum_{k=0}^n {{n}\choose{k}}2^k $$
However this doesn't really help.

This question is related to a famous unsolved problem in number theory: The distribution of the numbers $(3/2)^n$ modulo $1$. See here for a relatively recent paper that might contain additional references:  
https://arxiv.org/abs/1806.03559
If there were a simple answer to your question this problem (and another famous one) would be nearer to solution, or would even be solved by now.
