JavaScript (ES6), 26 bytes
Returns the \$n\$-th term of the sequence. Relies on arithmetic underflow to stop the recursion.
f=n=>n?(-n%4/3|1)*f(n/2):1
Returns the \$n\$-th term of the sequence. Relies on arithmetic underflow to stop the recursion.
f=n=>n?(-n%4/3|1)*f(n/2):1