Define 2 power series over the field $\mathbb Z/2\mathbb Z$ by $f=1+x+x^3+x^6+\dots$, the exponents being the triangular numbers, and $g=1+x+x^4+x^9+\dots$, the exponents being the squares. Write $f/g$ as $c_0+c_1x+c_2x^2+\dots$ with each $c_n$ in $\mathbb Z/2\mathbb Z$.

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 $\lbrace 0,6,10,28,36,\dots\rbrace$? Kevin O'Bryant has verified that this holds when $n$ is 512 or less.

Remark. If one writes $1/g$ as $b_0+b_1x+b_2x^2+\dots$, then $n\mapsto b_n$ is the characteristic function $\bmod 2$ of the set $B$ studied by O'Bryant, Cooper and Eichhorn (see this and this 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 $7 \bmod 16$.

(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 $B$).