Continued fractions arising from $${\mathcal F}_{1,3}$$ F 1 , 3

We study a family of continued fractions arising from a graph known as $${\mathcal F}_{1,3}$$ which is isomorphic to a subgraph of the Farey graph. We call these continued fractions $${\mathcal F}_{1,3}$$ -continued fractions. In fact, certain paths from infinity to a vertex in $${\mathcal F}_{1,3}$$ correspond to finite $${\mathcal F}_{1,3}$$ -continued fractions of the vertex and vice versa. Further, we study uniqueness of the longest $${\mathcal F}_{1,3}$$ -continued fraction expansions of real numbers and show that their convergents are best approximations of the numbers by vertices of $${\mathcal F}_{1,3}$$ .