In page 9 of Concrete Mathematics, regarding the Josephus Problem, they state that "each person's number has been doubled then decreased by 1".
$J(2n) = 2J(n) - 1$, for $n \ge 1$
I don't quite understand where the subtraction of 1 comes in.
This also goes for the odd problem where the formula is given as:
$J(2n+1) = 2J(n) + 1$, for $n \ge 1$
I know I'm missing an obvious piece of logic here but I haven't figured it out so far...