An alternating product/sum Fibonacci-like sequence
Hint: Study separately the sequence of even indices and odd indices. It's easy to see that for the sequence $S(\tfrac{1}{2},-\tfrac{1}{2})$ they satisfies:
The odd indices: $$o_1=\frac{1}{2},\;o_2=-\frac{1}{4},\; o_{n+2}=o_{n+1}^2\left(1+\frac{1}{o_n}\right)$$
The even indices:
$$e_1=-\frac{1}{2},\;e_2=-\frac{3}{4},\; e_{n+2}=e_{n+1}(e_{n+1}-e_n+1)$$