# 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 $$a+b$$.

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