Representation of powers of three in binary.

2018-10-13 10:57:47

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.