More about odd numbers in Pascal's triangle

Do you have any references to this interesting result? I could not find any...

The total number of odd numbers in the first $$2^n$$ rows of Pascal's triangle is $$3^n$$, $$n>=0$$.

It's easy to prove by induction based on the formula for the number of odds in row $$n$$ ($$2^m$$, where $$m$$ is the number of ones in the binary expansion of $$n$$).

• Just saw it: the answer by @Micah to the "Odds in Pascal's Triangle" post includes a direct proof of this fact. – Mircea Draghicescu Jan 29 at 22:10