Let $(a_n)_{n \geq 0}$ be the partial quotients of a [simple continued fraction](/page/Simple%20Continued%20Fraction). Define integer sequences $(p_n)_{n \geq -2}$ and $(q_n)_{n \geq -2}$ by $p_{-2}=0$, $p_{-1}=1$, $q_{-2}=1$, $q_{-1}=0$, and, for every $n \geq 0$, by the standard [convergent](/page/Convergent) recurrence $p_n=a_np_{n-1}+p_{n-2}$ and $q_n=a_nq_{n-1}+q_{n-2}$. Then, for every $n \geq 1$, the lattice vectors $(q_{n-1},p_{n-1})$ and $(q_n,p_n)$ satisfy $\det((q_{n-1},p_{n-1}),(q_n,p_n)) = (-1)^{n-1}$.