A Lucas Sum

Algebra Level 5

Let $p = \sqrt{1+\sqrt{1+\sqrt{1+\ldots}}}$ .

The sum

$\sum_{k=2}^{\infty} \frac{\lfloor p^{k} \rceil}{2^{k}}$

can be expressed as $\frac{a}{b}$ for $a,b$ coprime, where $\lfloor \cdot \rceil$ denotes the nearest integer function. Find $a+b$ .