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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 $a(F_n-1)$ always odd?

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    $\begingroup$ I seem to remember having done this problem as an exercise in Stanley's EC1. If memory serves correct a(n) is odd iff all the strings of consecutive zeros in the Zeckendorff representation of n are odd. $\endgroup$
    – Gjergji Zaimi
    Commented Dec 31, 2014 at 3:37
  • $\begingroup$ My guess about a(F_n-1) being odd was true for the first 21 values of n, but not true for larger values. $\endgroup$
    – David S. Newman
    Commented Dec 31, 2014 at 4:07
  • $\begingroup$ @DavidS.Newman I don't get numerical support for your claim. $\endgroup$
    – joro
    Commented Dec 31, 2014 at 7:04
  • $\begingroup$ Did you try using Proposition 1 of the paper you cite? For instance, its third conclusion says that $a(F_{n+1}-1)=a(F_{n-3}-1)$, and as your conjecture holds for first 4 values of $n$, it thus holds always (and moreover, you get an explicit formula -- signs are $-,-,+,+$ with a period of four). $\endgroup$
    – Victor Kleptsyn
    Commented Jan 1, 2015 at 17:34

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