Closed-form solution to recurrence equation
Define
$$
\begin{align*}
A(x) &amp;= \frac{1}{x^2} + 1 + x + x^3, \\
B(x) &amp;= \frac{x^3}{1-x^2} + \frac{x^2}{(1-x^2)^2} - \frac{1}{x^2} - 1 - 2x^2 \\ &amp;=
-\frac{1}{x^2} -1 - x^2 + \sum_{m=0}^\infty x^{3+2m} + \sum_{m=0}^\infty (2+m) x^{4+2m}.
\end{align*}
$$
Then the generating series $F(x) = \sum_{n=0}^\infty f(n) x^n$ satisfies the identity
$$ F(x) = A(x) F(x^2) + B(x), $$
and so in some sense is given by the "formula"
$$
F(x) = B(x) + A(x) B(x^2) + A(x) A(x^2) B(x^4) + A(x) A(x^2) A(x^4) B(x^8) + \cdots.
$$
Unfortunately, this formula holds only in the sense of analytic continuation.

If we use Dirichlet series instead of ordinary generating series, we can probably compute the asymptotics of the sequence, along the lines described here. We leave that as an exercise to the reader.