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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:

p(n) = f (n+3)

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:
f (n+1) = f (n) + f (n-1).

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