If $$f(n)=\left[\sqrt{n}+\frac{1}{2}\right]$$ where $[\cdot]$ denotes the greatest integer function, then find the value of the limit $$\lim\_{n\to\infty}\sum\_{k=1}^n\left(\frac{2^{f(k)}+2^{-f(k)}}{2^{n}}\right).$$

My attempt: I tried to evaluate individual values of $f(n)$ for different values of n to get a pattern, e.g. $$f(1) = f(2)= 1,\quad f(3) = f(4)= f(5) = f(6)= 2,\\ f(7) = f(8) = f(9) = f(10) = f(11) = f(12) = 3,\dots $$ 
but I feel there should be a generic way of handling this problem.

Any help will be appreciated.