Recurrence Equations Can someone explain how that Tn = 2Tn-1 + 1 sequence actually works? 

The subscript merely indicates which term of the sequence we are using. Thus, $T_n$ represents the value of the nth term of the sequence.  The above interpretation applies equally well to any other term (eg. $n - 1$). 
$T_n = 2T_{n-1} + 1$ means "to find the value of the nth term of the sequence, you need to find the value of $(n - 1)$th term first; then 2 times it; and then add 1 to it".
