Question 48
Let $\mathbb{N}_0$ denote the nonnegative integers. The function $f:\mathbb{N}_0\to\mathbb{N}_0$ satisfies $f(0)=0$ and
$$
f(2n)=2f(n),\qquad f(2n+1)=n+2f(n)\qquad(n\ge0).
$$
(a) Determine the sets
$$
\begin{aligned}
L&=\{n\in\mathbb{N}_0:f(n)\lt f(n+1)\},\\
E&=\{n\in\mathbb{N}_0:f(n)=f(n+1)\},\\
G&=\{n\in\mathbb{N}_0:f(n)\gt f(n+1)\}.
\end{aligned}
$$
(b) For each integer $k\ge0$, find a formula for
$$
a_k=\max\{f(n):n\in\mathbb{N}_0,\ 0\le n\le2^k\}.
$$
$$
f(2n)=2f(n),\qquad f(2n+1)=n+2f(n)\qquad(n\ge0).
$$
(a) Determine the sets
$$
\begin{aligned}
L&=\{n\in\mathbb{N}_0:f(n)\lt f(n+1)\},\\
E&=\{n\in\mathbb{N}_0:f(n)=f(n+1)\},\\
G&=\{n\in\mathbb{N}_0:f(n)\gt f(n+1)\}.
\end{aligned}
$$
(b) For each integer $k\ge0$, find a formula for
$$
a_k=\max\{f(n):n\in\mathbb{N}_0,\ 0\le n\le2^k\}.
$$