Recursion Question - Trying to understand the concept Just trying to grasp this concept and was hoping someone could help me a bit.  I am taking a discrete math class.  Can someone please explain this equation to me a bit?
$f(0) = 3$
$f(n+1) = 2f(n) + 3$
$f(1) = 2f(0) + 3 = 2 \cdot 3 + 3 = 9$
$f(2) = 2f(1) + 3 = 2 \cdot 9 + 3 = 21$
$f(3) = 2f(2) + 3 = 2 \cdot 21 + 3 = 45$
$f(4) = 2f(3) + 3 = 2 \cdot 45 + 3 = 93$
I do not see how they get the numbers to the right of the equals sign.  Please someone show me how $f(2) = 2f(1) + 3 = 2 \cdot 9 + 3$. I see they get "$2\cdot$" because of $2f$ but how and where does the $9$ come from?  I also see why the $+3$ at the end of each equation but how and where does that number in the middle come from?

Perhaps by considering a different sequence this may become clearer:
$$f(0)=0$$
$$f(n+1)=n+1$$
therefore
$$\begin{align}
f(n=1)&=f(n=0)+1=0+1=1\\
f(n=2)&=f(n=1)+1=(f(n=0)+1)+1=(0+1)+1=2\\
f(n=3)&=f(n=2)+1=(f(n=1)+1)+1=((f(n=0)+1)+1)+1=((0+1)+1)+1=3\\
\end{align}$$
So this will generate all the natural numbers.
