So given the sequence https://oeis.org/A227310 $a(n) = 1,1,0,0,1,0,0,1,0,1,1,0,2...$
with generationg function $\frac{1}{G(0)}$
with $G(k) = 1+ \frac{(-q)^{k+1}}{1-\frac{(-q)^{k+1}}{G(k+1)}}$
I see that the series expansion of the generating function is $1 + x + x^4 + x^7 ...$
But can the generating function be re-written as a sum or product that generates the coefficients directly?
Similar to say this generating function for the partition numbers.
$\prod_{k>0} 1/(1-x^k) $