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.