3.2.6.1 Lunda polyominoes and Lunda animalsPolyominoes (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 Lundaanimals, in his book Lunda geometry: Designs, Polyominoes, Patterns, Symmetries P. Gerdes (1996) obtained the first approximation for the total number of different Lunda nominoes. Lundaanimal is a Lunda momino with a unit square at one of its ends, representing a head. A Lundaanimal 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 Lundaanimal have a Lunda (m1)omino in common. How many different positions p_{5}(n) of a Lunda 5omino are possible after n steps? P. Gerdes proved that: p_{m}(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(n1). It is interesting that for every Lunda momino for m < 9 the result is the same, so p_{m}(n) = f(n+3) for 1 £m£8. From m = 9 onwards, p_{m}(n) < f(n+3). Open problem: try to find the general formula for p_{m}(n).
