The author generates a Tower of Hanoi and looks at the sequence:

$$1, 3, 7, 15, 31, 63,...$$

He guesses the recurrence relation from the first few terms:

$$H\_{n} = 2^{n} - 1$$

Now he wants to prove it using proof by induction:

> 
> **Base case**: For $n = 1$, it's true.
> 
> 
> **Induction case**: Assume for induction that $H\_{n - 1} = 2^{n - 1} - 1$, then
> 
> 
> $$\begin{align\*} H\_{n} & = 2H\_{n - 1} + 1 \\
> & = 2(2^{n - 1} - 1) + 1 \\
> & = 2^n - 1\end{align\*}
> $$
> 
> 
> 

I don't understand the above equation. First he says that $H\_{n} = 2^{n} - 1$ and then he changes it to $H\_{n} = 2H\_{n - 1} + 1$ - I don't get the logic in that.