Let $F_n$ denote the $n$th Fibonacci number.

Then $\prod\limits_{i=1}^{\infty}(1-x^{F_i})$ is a series all of whose coefficients are either $-1$, $0$ or $+1$.

The sequence of the coefficients in this series, denote them $a(n)$, is in the OEIS as A093996.

There is a reference there to a paper by Federico Ardila, giving a proof that the coefficients are all $-1$, $0$ or $+1$, as well as a number of recurrence relations for the coefficients.

I'm wondering if, using this information, it is possible to characterize some (or all) of the values in the sequence.

For example, is $a(F_n)$ always odd?