Lunda polyominoes and Lunda animals

Polyominoes (either black or white) appearing in Lunda designs will be called Lunda polyominoes (Gerdes, 1996). The possible shape of Lunda polyominoes is restricted by the local equilibrium condition for Lunda designs. Therefore, some polyominoes are inadmissible (e.g., 001001001001). On the other hand, Lunda polyominoes also include "hollow" polyominoes. By introducing the concept of Lunda-animals, in his book Lunda geometry: Designs, Polyominoes, Patterns, Symmetries P. Gerdes (1996) obtained the first approximation for the total number of different Lunda n-ominoes. 

Lunda-animal is a Lunda m-omino with a 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 every other will come into the position previously taken by the preceding cell. In other words, two subsequent positions of a Lunda-animal have a Lunda (m-1)-omino in common. How many different positions p5(n) of a Lunda 5-omino are possible after n steps? P. Gerdes proved that: pm(n) = f(n+3) for m = 1,2,3,...,8, 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(n) = f(n+3) for 1 £m£8. From m = 9 onwards, pm(n) < f(n+3). Open problem: try to find the general formula for pm(n).