Proof of the Modulo 4 Conjecture for OEIS A141117 and of the Parity Conjecture for OEIS A091713

Let 𝐹(𝑥) be the ordinary generating function of OEIS A091713, determined by 𝐹(𝑥) = 𝑥 + 𝑥𝐹◦3(𝑥),and let 𝐴(𝑥) = 𝐹◦2(𝑥), the ordinary generating function of OEIS A141117. We prove that every coefficientof 𝐹 is odd and that, for 𝑛 > 1, the coefficient 𝑎(𝑛) = [𝑥𝑛] 𝐴(𝑥) is divisible by 4 exactly when 3 | 𝑛, while𝑎(𝑛) ≡ 2 (mod 4) otherwise. The argument is entirely within rings of formal power series.1 Statement and conventionsLet 𝐹(𝑥) ∈ 𝑥Z[[𝑥]] be the ordinary generating function of OEIS A091713. Its defining fu