Subsets of {1,…,n} with Exactly One Pair of Consecutive Integers

Count the bit strings of length $n$ with exactly one pair of consecutive ones. The convolution $\sum F_i F_{n-i}$ equals the coefficient of $x^{n-2}$ in $1/(1-x-x^2)^2$; decomposing this by partial fractions over $r_\pm = (1\pm\sqrt 5)/2$ yields the closed form $f(n) = \tfrac{n-1}{5} L_n + \tfrac{2}{5} F_{n-1}$. Full derivation and computational verification in the PDF.