I'm not really familiar with number theory, or mathy stuff for that matter. However I recently noticed that if you take the even powers of 2 ( $2^n$ where $n$ is even) and write the difference of each two consecutive numbers ( $2^4 - 2^2$ which is 12), and turn those differences into a sequence and repeat, like so:
1, 4, 16, 64,256 3, 12, 48,192 9, 36,144 27,108 81 (let's call this 'm')
the final numbers is a power of 3.
Why does this happen? Is there any way to prove this? Is it a proven thing?