ISL 1996 A9

Let the sequence $a(n), n = 1,2,3, \ldots$ be generated as follows with $a(1) = 0,$ and for $n > 1:$

$$a(n) = a\left( \left \lfloor \frac{n}{2} \right \rfloor \right) + (-1)^{\frac{n(n+1)}{2}}.$$

1.) Determine the maximum and minimum value of $a(n)$ over $n \leq 1996$ and find all $n \leq 1996$ for which these extreme values are attained.

2.) How many terms $a(n), n \leq 1996,$ are equal to 0?