OliverA
Testing:
We will count the number of it $\< 2^{11}=2048$ (Error compiling LaTeX. Unknown error_msg) instead of (In other words, the length of the base-2 representation is at most
. If there are even digits,
, then the leftmost digit is
, the rest,
, has odd number of digits. In order for the base-2 representation to have more
's, we will need more
in the remaining
than
's. Using symmetry, this is equal to
Using similar argument where there are odd amount of digits. The remaining even amount of digit must contains the number of
's at least as the number of
's. So it's equal to
Summing both cases, we have
. There are
numbers between
and
inclusive that satisfy it. So the answer is