- The number of different possible games is F
_{n}= (Phi^{n}− phi^{n}) /, where F_{n}is the nth Fibonacci number, defined by the recurrence equation F_{1}= 1, F_{2}= 1, F_{k}= F_{k−1}+ F_{k−2}, for k > 2, and Phi = (1 + )/2 and phi = (1 − )/2 are the roots of the quadratic equation x^{2}− x − 1 = 0.. - The number of possible games of length k is given by , where k must have the same parity as n.