The Stern-Brocot continued fraction (Q2926264)

Let \(s_{0,1} =0\) and \(s_{0,2} =1\). For \(n \geq 1\), the sequence \(S_n=\langle s_{n,1},s_{n,2},\dots , s_{n,2^n+1}\rangle\) is defined via the rule \( s_{n,2k-1}=s_{n-1,k} ,\;s_{n,2k}=s_{n-1,k}+s_{n-1,k+1}\) for \(k\geq 1\). Starting with \(q_{0,1} =1\) and \(q_{0,2} =0\), we define the sequence \(Q_n=\langle q_{n,1},q_{n,2},\dots , q_{n,2^n+1}\rangle\) via the same rule as for \(S_n\). The Stern-Brocot sequence of order \(n\) for \(n\geq 1\) is the sequence \(H_n=\langle h_{n,1},h_{n,2},\dots , h_{n,2^n+1}\rangle\) defined via the rule \( h_{n,i}=s_{n,i}/q_{n,i}.\) Apparently, the Stern-Brocot tree has deep applications to physical chemistry. There are many known properties of the Stern-Brocot sequence.NEWLINENEWLINEIn the paper under review, Bates gives many new interesting properties, which particularly involve continued fractions. More precisely, let us quote the author: ``We discover a continued fraction whose successive approximants generate the Stern-Brocot sequence and levels of the Stern-Brocot tree. We also discover continued fractions whose approximants generate every term in diagonals and branches of the Stern-Brocot tree''.