The Tower of Hanoi problem:

According to legend, a certain Hindu temple contains three thin diamond poles($\displaystyle A$, $\displaystyle B$, $\displaystyle C$ and $\displaystyle A$ is closer to $\displaystyle B$ and $\displaystyle B$ is closer to $\displaystyle C$ but $\displaystyle C$ is not closer to $\displaystyle A$) on one of which, at the time of creation, God placed $\displaystyle 64$ golden disks that decrease in size as they rise from the base.

The priests of the temple work unceasingly to transfer all the disks one by one from the first pole to one of the others, but they must never place a large disk on top of a smaller one and they are allowed to move disks from one tower to only adjacent pole. Let

$\displaystyle a_n = \left[ \begin{array}{c} \text{the minimum number of moves} \\ \text{needed to transfer a tower of n} \\ \text{ disks from pole A to pole C} \end{array} \right ] $

$\displaystyle b_n = \left[ \begin{array}{c} \text{the minimum number of moves} \\ \text{needed to transfer a tower of n} \\ \text{ disks from pole A to pole B} \end{array} \right ] $

Find a recurrence relation $\displaystyle a_n$ and $\displaystyle b_n$.

Solve for the problem to find $\displaystyle a_n$ (The book did this):

$\displaystyle \begin{align*}\nonumber a_n =& a_{n-1} \text{ (moves to move the top n - 1 disks from pole A to pole C) } +\\ &\nonumber 1 \text{ (move to move the bottom disk from pole A to pole B) } + \\ & \nonumber a_{n-1} \text{ (moves to move the top disks from pole C to pole A) } + \\ & \nonumber 1 \text{ (move to move the bottom disk from pole B to pole C) } + \\ & \nonumber a_{n-1} \text{ (moves to move the top disks from pole A to pole C) } \\ \nonumber =& 3a_{n-1} + 2 \end{align*} $

Solve for the problem to find $\displaystyle b_n$(I did this):

$\displaystyle \begin{align*}\nonumber b_n =& b_{n-1} \text{ (moves to move the top n - 1 disks from pole A to pole C) } +\\ & \nonumber 1 \text{ (move to move the bottom disk from pole A to pole B } + \\ & \nonumber b_{n-1} \text{ (moves to move the top disks from pole C to pole B) } \\ \nonumber =& 2b_{n-1} + 1 \end{align} $

Why my solution of $\displaystyle b_n$ is wrong? For finding $\displaystyle b_n$ I did the same thing as done in finding $\displaystyle a_n$ which is skipping the move from $\displaystyle A$ to $\displaystyle B$ when moving from $\displaystyle A$ to $\displaystyle C$. I calculated this move as $\displaystyle 1$ like the book did it.

So why am I wrong?

Also my question is why in finding $\displaystyle a_n$ to move the disks from $\displaystyle A$ to $\displaystyle C$ they skipped to calculate the move to move the disks first from $\displaystyle A$ to $\displaystyle B$?