Suppose a bicycle full a place in a park of bycicles and a motorcycle full two places, find a recurrence relation that give us the number of possible ways to park bicycles and motorcycles in a park with $n$ places.
If the park have $0$ places, then, $a_0=0$,
if the park have $1$ place, then, $a_1=1$,
if the park have $2$ places, then, $a_2=2$
if the park have $3$ places, then, $a_3=2$....
The first $11$ terms in the relation are: $$0,1,2,2,3,4,4,5,5,6,6$$
But, how i can make the relation of recurrence if there are terms that do not depend on the previous term.
Thanks.
You seem only to care about the number of bicycles and motorcycles, not the order they are parked in. The wording of the question seems to me to care about order, so for three places you have BBB, BM, MB for three possibilities.
If you only care about how many of each type, you just have to specify the number of motorcycles. This can range from $0$ to $\lfloor \frac n2 \rfloor$, so there are $1+\lfloor \frac n2 \rfloor$ combinations.
The simple to understand recurrence is if you add two spaces you add the capacity for one more motorcycle, so $a_n=1+a_{n-2}$