Suppose that you have a large supply of red, white, green and blue poker chips. you want to make a vertical stack of n chips in such a way that the stack does not contain any consecutive blue chips.
Find a recurrence relation for $a_n$ where $ a_n$ denotes the number of ways you can make such a stack of n poker chips.
Blue chip first then i have 3 choices for next chip of the n-2 remaining symbols i write ($ a_{n-2} $)?
for the second case where blue is not the first chip simply let ( $a_{n-1} $ ) be the Operation? symbol? function? i dont know on the remaining n-1 symbols.
$a_n = 3( a_{n-1} )+ 3( a_{n-2} ) $
Does this result make any sense?
Just add some helpful definitions:
Define $r_n$ as any sequence ending in red, white, or green.
Define $b_n$ as any sequence ending in blue.
Obviously $a_n = b_n + r_n$, and your reasoning to find $r_n$ and $b_n$ is correct.