$p = \sqrt{1+\sqrt{1+\sqrt{1 + \cdots}}}$; $\sum_{k=2}^{\infty}{\frac{\lfloor p^k \rceil}{2^k}} = ? $ 
Let $p = \sqrt{1+\sqrt{1+\sqrt{1 + \cdots}}}$
  The sum $$\sum_{k=2}^{\infty}{\dfrac{\lfloor p^k \rceil}{2^k}}$$
  Can be expressed as $\frac{a}{b}$. Where $\lfloor \cdot \rceil$ denotes the nearest integer function. Find $a+b$.  

My Work
$p^2 = 1+p \implies p = \dfrac{1+\sqrt{5}}{2} = \phi$
And also $\phi^n = \dfrac{L_n + F_n\sqrt{5}}{2}$. Where $L_n$ is the $n-th$ Lucas-Number.
What to do with the next part?    
Note: Problem Collected from Brilliant.org 
 A: You are correct that $$p=\frac{1+\sqrt{5}}{2}$$
This is the golden ratio, ususally labeled $\phi$. There is a well-known relationship between $\phi$ and the Lucas numbers $L_k$: for all natural numbers $k$, 
$$L_k = \phi^k + \left(\frac{-1}{\phi}\right)^k$$
The second term above will be less than $1/2$ for $k >1$, so rounding $\phi^k$ to the nearest integer must give $L_k$.

This means your sum can be written in terms of Lucas numbers. Let your sum be $S$. Then
$$S = \displaystyle\sum\limits_{k=2}^{\infty} \frac{L_k}{2^k}$$
Now we can write out a few terms of $S$, $2S$, and $4S$:
\begin{align}
S = \frac{L_2}{4}&+\frac{L_3}{8}+\frac{L_4}{16}+\frac{L_5}{32}+\cdots \\\\
2S = \frac{L_2}{2}+\frac{L_3}{4}&+\frac{L_4}{8}+\frac{L_5}{16}+\cdots \\\\
4S = L_2+\frac{L_3}{2}+\frac{L_4}{4}&+\frac{L_5}{8}+\frac{L_6}{16}\cdots \\
\end{align}
I've intentionally lined up the sums above like that because if we add up the first two equations, term by term, we get
$$
3S = \frac{L_2}{2} + \frac{L_2+L_3}{4} + \frac{L_3+L_4}{8}+\frac{L_4 + L_5}{16}+\cdots
$$
But the Lucas numbers, like the Fibonacci numbers, satisfy the recursion $L_k + L_{k+1} = L_{k+2}$. Therefore 
$$3S = \frac{L_2}{2} + \left(\frac{L_4}{4} + \frac{L_5}{8} + \frac{L_6}{16}+\cdots\right)$$
But everything after the first term looks a lot like what we had written for $4S$ above! In fact, we can write
$$3S = \frac{L_2}{2} + \left(4S - L_2 - \frac{L_3}{2}\right)$$
$$S = \frac{L_2}{2} + \frac{L_3}{2}$$
$$S = \frac{L_4}{2}$$
Finally, we can solve for your sum: 
$$S = \frac{7}{2}$$
A: For $n \ge 2$, the numbers $\lfloor \phi^k \rceil$ are equal to the Lucas numbers, which satisfy the Fibonacci recurrence but in which $L_1=1$ and $L_2=3$.  The Lucas numbers satisfy
$$L_k = \phi^k + (-1)^k \phi^{-k}$$
and thus have the generating function
$$L(x) = \frac{2-x}{1-x-x^2} $$
so the sum we want is
$$L \left (\frac12 \right ) - L_0 - \frac12 L_1 = \frac72$$ 
Note that $\phi^{-1}= \phi-1 = 0.618\ldots$ and exponentiation to the power of $k$ does not affect a rounding for $k \ge 2$.  
