TITLE: Why does this sequence happen like this?
QUESTION [6 upvotes]: The other day I sent my girlfriend this text
<3
she sent me back
<3<3<3
not to be one upped I responded with
<3<3<3<3<3<3<3
this got very silly very quickly.
After our "<3" battle was over I got to thinking about the pattern we were forming. Since we were doubling the number of hearts and adding one I thought the sequence would be something like 2^n + 1. BUT IT IS NOT. It is 2^n - 1 .
Why is this?

REPLY [5 votes]: Because you started with $1=2^1-1$.  Then $2*1+1=2*(2^1-1)+1=2^2-1$.  Generally $2(2^n-1)+1=2^{n+1}-1$  When you double the $-1$ and add $1$, you get $-1$ back again.

REPLY [2 votes]: The pattern clearly holds for the first few terms. Let's see if going from step $n$ to step $n+1$ keeps with the pattern.
$a_n=2^n-1$. Then
$$
a_{n+1}=2a_n+1=2(2^n-1)+1=2^{n+1}-2+1=2^{n+1}-1.
$$