Lunda-animal is a Lunda 5-omino with one unit square at one of its ends representing a head. A Lunda-animal walks in such a way that after each step the head occupies a new unit square, and each other cell occupies the preceeding one. In other words, two subsequent positions of a Lunda-animal have a Lunda-tetromino in common. 

How many different positions p₅(n) of a Lunda 5-omino are possible after n steps? 
P.Gerdes proved that: 

for n=1,2,3,..., where f (n) is the famous Fibonacci sequence 0,1,1,2,3,5,8,13,21,34... given by the recurrence formula: 

It is interesting that for every Lunda m-omino for m<9 the result is the same, so p_m = f (n+3) for 4<m<9. From m=9 onwards, p_m < f (n+3). Try to find the general formula for p_m!