Melodic Series

If $a_1$, $a_2$, $a_3$, ... is an Artistic Series, then $1/a_1$, $1/a_2$, $1/a_3$, ... is a Melodic Series. The Recurrence Relation obeyed by melodic series is

$$b_{i+3}={b_i{b_{i+2}}^2\over{b_{i+1}}^2}+{{b_{i+2}}^2\over b_{i+1}}-b_{i+2}.$$

See also Artistic Series

References

Duffin, R. J. ``On Seeing Progressions of Constant Cross Ratio.'' Amer. Math. Monthly 100, 38-47, 1993.