Answer by Cesareo for solve recurrence relation $a_{n+2}=(a_{n+1}+1)/a_n$

Calling $a_1 = c_1$ and $a_2 = c_2$ we have

$$\cases{a_1 = c_1\\a_2 = c_2\\a_3 = \frac{c_2+1}{c_1}\\a_4 = \frac{c_1+c_2+1}{c_1c_2}\\a_5 = \frac{c_1+1}{c_2}\\a_6 = c_1\\a_7 = c_2\\\vdots}$$