Odd behavior in the coefficients of reciprocals of binary power series

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Let A be a finite subset of N including 0 and fₐ(n) be the number of ways to write n=∑ᵢ₌₀∞εᵢ2ⁱ, where εᵢ. The sequence (fₐ(n)) 2 is always periodic, and fₐ(n) is typically more often even than odd. We give four families of sets (Aₘ) with |Aₘ|=4 such that the proportion of odd fₐₘ(n)'s goes to 1 as m→∞.