Define 2 power series over Z/2 by f=1+x+x^3+x^6+..., the exponents being the triangular numbers, and g=1+x+x^4+x^9+..., the exponents being the squares. Write f/g as c_0+(c_1)x+
(c_2)x^2+... 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...}? 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+..., then n-->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).