# Being even or odd in the product expansion $\prod(1+x^k+x^{k+1})$

Consider the generating function of "partitions with distinct parts" $$\sum_nQ(n)x^n=\prod_k(1+x^k).$$ It's known that $$\left[\prod_k(1+x^k)\right] \mod 2=\prod_m(1-x^m)=\sum_{j\in\mathbb{Z}}x^{j(3j-1)/2}.$$ So, $$Q(n)$$ is "more often" even than odd.

Define the product $$\prod_{k\geq1}(1+x^k+x^{k+1})=\sum_nU(n)x^n$$ and the proportionals $$\eta(N)=\frac{\#\{0\leq n\leq N: \text{U(n) is odd}\}}N.$$

QUESTION. Is it true that $$\eta(N)\geq\frac12$$, for large $$N$$? Experiments seem to suggest so.

• for the record, this is oeis.org/A160571 – Carlo Beenakker Apr 17 at 19:52
• I've computed $U(n)$ mod $2$ up to $n=100000$ and it doesn't seem true. We have $\eta(N)\ge \frac{1}{2}$ up to roughly $N=4000$ but at this point $\eta(N)$ goes below $\frac{1}{2}$ and stays there for a while. $\eta(N)$ crosses $\frac{1}{2}$ again around $N=99500$. The behavior of $\eta(N)$ seem overall somewhat chaotic and I think that it might well cross $\frac{1}{2}$ infinitely often. – Antoine Labelle May 9 at 19:27