IMC 2013 Problem 10

Consider a circular necklace with ${2013}$ beads. Each bead can be paintes either green or white. A painting of the necklace is called good if among any ${21}$ successive beads there is at least one green bead. Prove that the number of good paintings of the necklace is odd.Note. Two paintings that differ on some beads, but can be obtained from each other by rotating or flipping the necklace, are counted as different paintings.

Proposed by Vsevolod Bykov and Oleksandr Rybak, Kiev.