I'm working my way though a section on Second-Order Linear Homogenous Recurrence Relations with Constant Coefficients. There is an example that I do not understand. The part I'm having trouble with is highlighted in red.
Why does $n\in A$ mean that $A - {n}$ would be counted in $a_{n-2}$. Part b makes sense because the biggest possible element in A is then $n-1$ which would be counted in $a_{n-1}$ if is smaller than that, it would still be counted in $a_{n-1}$. These are the only two possible cases so acceptance of a) is all I'm really missing to complete my understanding. Can anyone help me understand why a) is true?
If $n \in A$, then $n-1 \not\in A$ (otherwise, $A$ will contain consecutive integers $n-1$ and $n$). Thus maximum possible value in $A-\{n\}$ is $n-2$ and $A-\{n\}$ can now be chosen in $a_{n-2}$ ways.