Question 147
Let $f$ be a function from nonnegative integers to nonnegative integers such that $f(0) = 0$ and
$$
f(m) = f\left(\left\lfloor \frac{m}{2} \right\rfloor\right) + \left\lceil \frac{m}{2} \right\rceil^2
$$
for all positive integers $m$. Compute
$$
\frac{f(1)}{1 \cdot 2} + \frac{f(2)}{2 \cdot 3} + \frac{f(3)}{3 \cdot 4} + \dots + \frac{f(31)}{31 \cdot 32}.
$$
(Here, $\lfloor z\rfloor$ is the greatest integer less than or equal to $z$, and $\lceil z\rceil$ is the least positive integer greater than or equal to $z$.)
$$
f(m) = f\left(\left\lfloor \frac{m}{2} \right\rfloor\right) + \left\lceil \frac{m}{2} \right\rceil^2
$$
for all positive integers $m$. Compute
$$
\frac{f(1)}{1 \cdot 2} + \frac{f(2)}{2 \cdot 3} + \frac{f(3)}{3 \cdot 4} + \dots + \frac{f(31)}{31 \cdot 32}.
$$
(Here, $\lfloor z\rfloor$ is the greatest integer less than or equal to $z$, and $\lceil z\rceil$ is the least positive integer greater than or equal to $z$.)