Define 2 power series over $Z/2$ by $f=1+x+x^3+x^6+\cdots$, the exponents being the triangular numbers, and $g=1+x+x^4+x^9+\cdots$, the exponents being the squares. Write $f/g$ as $c_0+c_1x+c_2 x^2+\cdots$ with each $c_n$ in $Z/2$.

Question: Is it true that when $n$ is even then $c_n$ is $1$ precisely when $n$ is in the set of even triangular numbers $\{0,6,10,28,36,\ldots\}$? Kevin O'Bryant has verified that this holds when $n$ is 512 or less.

Remark: If one writes $1/g$ as $b_0+b_1 x+b_2 x^2+\cdots$, then $n\mapsto b_n$ is the characteristic function mod 2 of the set $B$ studied by O'Bryant, Cooper and Eichhorn (see two questions of O'Bryant on MO); they show that when $n$ is even then $b_n$ is 1 precisely when $n$ is twice a square. A positive answer to my question would give a nice characterization of those elements of $B$ that are congruent to 7 mod 16.

(I've used the modular forms tag because of the formal similarity of $f$ and $g$ to Jacobi theta functions, and the motivation of O'Bryant, Cooper and Eichhorn in looking at $B$).