I am trying to find some sort of pattern in the base-$2$ representation of $3^n$; in particular, I would like to find formulae for the number of ones in the binary representation of $3^n$, or at least some good bounds. This sequence is in the OEIS, but no useful information about it can be found there.
As for a strategy - my initial thoughts are that I should try and use the same technique that John Conway used when he analyzed the “Look-and-say” sequence by noticing that certain substrings of the digit sequences of numbers in the sequence fell into predictable orbits. However, this idea has not panned out so far.
Can anyone offer any interesting observations that will shine light on this problem?
Once $n$ gets large you can get a reasonable approximation by assuming the bits (aside from the first and last, but they are a small fraction of the total) are equally likely to be $0$ or $1$. $3^n$ in binary has $n \log_2 3 \approx 1.585n$ bits. You can use the normal approximation to get an idea of the dispersion around this.