Joe eats bananas, oranges, apples and strawberries every day. Joe would never eat 2 bananas or 2 oranges in a row, and after eating an apple he would only eat apples. Let $f(n)$ be the number of ways Joe can eat $n$ fruits a day (the order of the fruits matters).
I need to find the recurrence relation for $f(n)$.
I tried to separate the cases according to the last fruit he ate,
so if it was an apple there are $f(n-1)$ options,
if it was a strawberry there are $f(n-1)-f(n-2)$ options (because there are $f(n-1)$ options without any limitation and the we have to subtract the cases that he ate an apple before the strawberry which are $f(n-2)$) ,
I am stuck about the banana and similarly for the orange.
Would appreciate any help how to proceed:)