This sequence is present at OEIS A171503.
There, Jacob Siehler explains how this sequence correspond to the number of matrices with determinant one and how this number grows as we allow to vary in the set $\{0,1,...,n\}$ of available labels. Its first few terms are: $$3,7,15,23,39,47,71,87,111,127,...$$
There the recursion in the title is told.
Further, it is easy to verify that the sequence also counts the number of rational numbers within the labels $\{0,1,...,n\}$ varies:
\begin{eqnarray*} \{0,1\}&--& 0,\pm1/1\\ \\ \{0,1,2\}&--& 0,\pm1/1,\pm2,\pm1/2\\ \\ \{0,1,2,3\}&--& 0,\pm1/1,\pm2,\pm1/2,\\ &&\pm3,\pm1/3,\pm2/3,\pm3/2\\ \\ \{0,1,2,3,4\}&--& 0,\pm1/1,\pm2,\pm1/2,\pm3,\pm1/3,\pm2/3,\pm3/2,\\ &&\pm4,\pm1/4,\pm3/4,\pm4/3\\ \\ \{0,1,2,3,4,5\}&--& 0,\pm1/1,\pm2,\pm1/2,\pm3,\pm1/3,\pm2/3,\pm3/2,\pm4,\pm1/4,\pm3/4,\pm4/3,\\ &&\pm5,\pm1/5,\pm2/5,\pm3/5, \pm4/5,\pm5/2,\pm5/3,\pm5/4 \end{eqnarray*} Evindently we are using of the Eulerphi's function.
In sight of this, and with the aim of increasing the folklore, I am asking if in ours audience (MSE) someone else knows or had found others relations or connections with this interesant sequence.