Show that $f_{n-1} + L_n = 2f_{n}$.
So we need to find a $2$ to $1$ correspondence.
Set 1: Tilings an $n$-board.
Set 2: Tiling of an $n-1$-board or tiling of an $n$-bracelet.
So we need to decompose a tiling of an $n$-board to a tiling of an $n-1$-board or a tiling of an $n-1$-bracelet?
Source: Proofs that Really Count by Art Benjamin and Jennifer Quinn