they seem to be always close to the double of the previous value, excluding the first few values.
There may, or may not be an actual pattern to the values, but that "close to the double of the previous value" would occur if you simply chose a
completely random number from a set with 1 additional binary digit at each level.
Effectively each new private key would be a number of binary digits equivalent to the number of milibitcoins stored at the address with the first digit being a 1.
Let us assume that there is some sort of mathematical formula to solve this puzzle.
That's the only way we will ever be able to solve this puzzle during our lifetime.
I strongly recommend to all the mathematical experts out there to have a look at this sequence, maybe they are able to find something.