Question 4
Consider $f:\mathbb N_0\to\mathbb N_0$, where $\mathbb N_0$ is the set of nonnegative integers, defined by
$$f(0)=0,\qquad f(2n)=2f(n),\qquad f(2n+1)=n+2f(n)\quad(n\ge0).$$
(a) Determine $L=\{n:f(n)\lt f(n+1)\}$, $E=\{n:f(n)=f(n+1)\}$ and $G=\{n:f(n)\gt f(n+1)\}$.
(b) For every $k\ge0$, find a formula for $a_k=\max\{f(n):0\le n\le2^k\}$.
$$f(0)=0,\qquad f(2n)=2f(n),\qquad f(2n+1)=n+2f(n)\quad(n\ge0).$$
(a) Determine $L=\{n:f(n)\lt f(n+1)\}$, $E=\{n:f(n)=f(n+1)\}$ and $G=\{n:f(n)\gt f(n+1)\}$.
(b) For every $k\ge0$, find a formula for $a_k=\max\{f(n):0\le n\le2^k\}$.