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)$$a(F_n-1)$ always odd?