Problem: For each positive integer $n$,
let $f(n)$ denote the number of ways of representing $n$
as a sum of powers of 2 with nonnegative integer exponents.
Representations which differ only in the ordering
of their summands are considered to be the same.
For instance, $f(4) = 4$,
because the number $4$ can be represented in the following four ways:
$4$; $2+2$; $2+1+1$; $1+1+1+1$.

Prove that for any integer $n \geq 3$
we have $2^{\frac{n^2}{4}} < f(2^n) < 2^{\frac{n^2}2}$.

\end{enumerate}

Solution: It's clear that $f$ is non-decreasing.
By sorting by the number of $1$'s we used,
we have the equation
\[ f(N) =
  f\left( \left\lfloor \frac N2 \right\rfloor \right)
  +  f\left( \left\lfloor \frac N2 \right\rfloor -1 \right)
  +  f\left( \left\lfloor \frac N2 \right\rfloor -2 \right)
  + \dots
  + f(1) + f(0). \quad (\bigstar)
\]

\paragraph{Upper bound.}
We now prove the upper bound by induction.
Indeed, the base case is trivial and for the inductive step
we simply use $(\bigstar)$:
\[ f(2^n) = f(2^{n-1}) + f(2^{n-1}-1) + \dots
  < 2^{n-1} f(2^{n-1})
  < 2^{n-1} \cdot 2^{\frac{(n-1)^2}{2}}
  = 2^{\frac{n^2}{2} - \half}.
\]

\paragraph{Lower bound.}
First, we contend that $f$ is convex.
We'll first prove this in the even case
to save ourselves some annoyance:
\begin{claim*}
  [$f$ is basically convex]
  If $2 \mid a+b$ then
  we have $f(2a) + f(2b) \ge 2 f\left( a+b \right)$.
\end{claim*}
\begin{proof}
  Since $f(2k+1) = f(2k)$, we will only prove the first equation.
  Assume WLOG $a \ge b$ and use
  $(\bigstar)$ on all three $f$ expressions here;
  after subtracting repeated terms, the inequality then rewrites as
  \[ \sum_{(a+b)/2 \le x \le a} f(x)
    \ge \sum_{b \le x \le (a+b)/2} f(x). \]
  This is true since there are an equal number of terms on each side
  and $f$ is nondecreasing.
\end{proof}
\begin{claim*}
  For each $1 \le k < 2^{n-1}$, we have
  \[ f(2^{n-1} - k) + f(k+1) \ge 2f(2^{n-2}) \]
\end{claim*}
\begin{proof}
  Use the fact that $f(2t+1)=f(2t)$ for all $t$
  and then apply convexity as above.
\end{proof}
Now we can carry out the induction:
\[ f(2^n) = f(2^{n-1}) + f(2^{n-1}-1) + \dots
  > 2^{n-1} f(2^{n-2}) + f(0)
  > 2^{n-1} 2^{\frac{(n-2)^2}{4}} = 2^{\frac{n^2}{4}}.
\]