Artistic Series

A Series is called artistic if every three consecutive terms have a common three-way ratio

$$P[a_i, a_{i+1}, a_{i+2}]={(a_i+a_{i+1}+a_{i+2})a_{i+1}\over a_i a_{i+2}}.$$

A Series is also artistic Iff its Bias is a constant. A Geometric Series with Ratio $r>0$ is an artistic series with

$$P={1\over r}+1+r\geq 3.$$

See also Bias (Series), Geometric Series, Melodic Series

References

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