Yes, there are many very old (1850s) and very arbitrary formulas. Eric Renner's book lists a few and gives some historic background. I like his book for its creative content and the inspiration it provides, but it does not go too deep into the mathematics of pinhole imaging.

Recent research and MTF studies at the Royal Institute of Technology in Stockholm, Sweden have shown that there are only two equations with a solid mathematical foundation. One for max resolution (based on Rayleigh criterion), and one for max sharpness or contrast (based on Airy disc). You can safely ignore all others as they all lie somewhere between the two equations or are just in a different format. Since most people value sharpness over resolution, I based the above table on the optimum pinhole diameter for max sharpness, which is based on the Airy disc:

d = SQRT (2.44 * wavelength * focal length)

or sometimes shown as:

d = 1.56 * SQRT (wavelength * focal length)

As to the wavelength, the visible spectrum ranges from roughly 400-700 nm and film is sensitive to the most part of it. Plugging these values into the equation gives quite a spread (almost 200%). Unfortunately, you have to pick a value, because pinholes cannot be corrected for chromatic aberration. It makes sense to pick a medium value at which the eye is most sensitive and push the threshold of sharpness to the boundaries of the visible spectrum. Hence the suggestion of 550 nm.

Pinhole shape has a big influence. You can use that creatively (see Renner's book) or you can go for a laser-cut pinhole to optimize image quality. I can offer a contact at the University of Munich, Germany who can laser-cut pinholes into thin brass plates for you at a reasonable price. he made all of mine.