Answer to puzzle 146: Odds and evens

• The number of different possible games is F_n = (Phiⁿ − phiⁿ) /, where F_n is the nth Fibonacci number, defined by the recurrence equation F₁ = 1, F₂ = 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 x² − x − 1 = 0..
• The number of possible games of length k is given by , where k must have the same parity as n.