I am trying to convert the following recursive function to a non-recursive equation:
$$ f(n) = \begin{cases} 0,&\text{if n = 0;}\newline 2 \times f(n -1) + 1,&\text{otherwise.} \end{cases} $$
I have calculated the results for $n = 1$ through to $n = 6$ but I cannot find a discernible pattern from which to make an equation. The results are:
$$f(1) = 1$$
$$f(2) = 3$$
$$f(3) = 7$$
$$f(4) = 15$$
$$f(5) = 31$$
$$f(6) = 63$$
I hope I have formatted this correctly (my first time using LaTex), and if anyone could offer any help that would be greatly appreciated.