According to this video:
The Secrets of the Solar System - The Fibonacci Sequence & Orbital Resonances | Andromeda
our solar system’s planetary orbital periods seem to coincide to simple ratios.
(not perfectly, but close enough to make some interesting spirographs 
)
Could an analysis of orbital resonances theoretically point to the location of Planet 9?
That’s actually what you get from an extreme lack of resonance. Take any two quantities of the same units in nature, and their ratio will almost always be irrational (that is to say, it’ll always be irrational, unless there’s some good reason for it not to be). But any irrational number can be approximated by a rational number, or a ratio of two integers.
If two planets have orbital periods that are a “simple” rational number (that is, a ratio of two relatively small integers), you’ll get a resonance. Resonances in orbital periods tend to lead to instability. So if you have a solar system where two planets have a resonance, over time, it’ll generally evolve into a different state, where they don’t have a resonance. Even if the ratio isn’t a true rational number, but is merely very close to a “simple” rational number, you’ll get a similar effect. For instance, pi is very close to 22/7. So you’d expect ratios of orbital periods to be about as far as you can get from being approximated by simple rational numbers.
Well, the irrational number that’s as far as possible from being approximated by simple rational numbers is \phi, the famous Golden Ratio. So we’d expect to see the Golden Ratio showing up in ratios of the planets’ orbital periods. And if you attempt to approximate the Golden Ratio using rational numbers anyway (say, get the best approximation you can get with a denominator less than some number), the approximations you always get are the ratio of two successive Fibonacci numbers.
This is also the reason why the Fibonacci numbers show up elsewhere in nature, incidentally. For instance, if you’re a sunflower, and arranging your seeds around the middle of a flower, you don’t want anything like each seed being 90º around the flower from the previous one, because that’d result in the seeds just forming a big cross, with a lot of wasted space. You want the angle between one seed and the next to be an irrational fraction of the way around the circle, so they don’t pile up. And you want it to be a “very irrational” fraction, or they’ll still clump up more than you’d like. So the angle from one seed to the next is also the Golden Ratio times the angle around a full circle, and that also leads to the Fibonacci numbers.
The law that gives you the spacing of the planets was originally the Titius-Bode law (which predicted where the asteroids ought to be).
In the 20th century Blagg and Richardson independently refined the law, fitting it not only to the planets in orbit around the sun, but for the moons around the planets. the Blagg-Richrdson law works surprisingly well.
You can read more than you ever wanted to know about these laws here:
As phrased, this sounds very much like teleology… 
But I’m fairly sure you didn’t mean it like that, and it could be rephrased from a Darwinian perspective…
I mean, in a literal sense, a sunflower doesn’t “want” anything. But it’s a lot easier to say “want” than “derives a selective advantage from, such that the plants that do it this way have more descendants on average”.
Very true. Unfortunately an awful lot of people still don’t ‘get’ evolution… 