Converging Sums of a Fractal Sequence
Python 2, 55 49 42
I have no idea what's going on, but it seems hard to beat the Maple formula from the OEIS page.
This uses 0-based indexing.
f=lambda n,t=0:n&lt;1or f(n/2,n%2)-~-t*f(n-1)

Thanks to @PeterTaylor for -6 bytes.