Rydberg's formula

In the latter part of the nineteenth century, a German school teacher by the name of Rydberg made up an empirical formula that gave the wavelengths of the emissions of a hydrogen atom.

...where n and m are integers.

When n = 2 and m = 1, 2, 3, 4, etc. the formula gives the Balmer series.

The reason for the success of Rydberg's slightly odd formula was not known at the time, but remarkably, as wavelength measurements were extended into the ultraviolet and then into the infrared, the formula gave each of the five sequences with n increasing from 1 for the Lyman series in the UV, through 2 for the Balmer series, and 3, 4 and 5 for series at progressively longer wavelengths in the infrared.

Bhor realized that any new theory of the hydrogen atom had to somehow reproduce Rydberg's formula.

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