Continued fraction : $\zeta(3)=[a_0,a_1,a_2,a_3,\dots]=[1,4,1,18,1,1,1,4,1,9,\dots]$

 $k$-th approximation $\frac{p_k}{q_k}$ :  $\frac{p_0}{q_0}<\frac{p_2}{q_2}<\cdots<\zeta(3)<\cdots<\frac{p_3}{q_3}<\frac{p_1}{q_1}$

 $p_k=a_k p_{k-1}+p_{k-2},$ $p_{-1}=1,p_{-2}=0,$  $q_k=a_k q_{k-1}+q_{k-2},$ $q_{-1}=0,q_{-2}=1$

 See Paper by Prof.Hirata lab (J)