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In addition to the other replies, n=3 is also the point at which the system becomes chaotic. AFAIK only a handful stable configurations for three bodies in orbit are known; most initial conditions are difficult (impossible?) to solve for analytically. It is a classic example in chaos theory.

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There are quite a lot of known solutions now (12k): https://arxiv.org/abs/2308.16159

AFAIK figure-8 is only stable configuration. others config like euler, lagrange configuration is can be easily broke by small disturbances.

figure-8 trajectory is very hard to calculate for me. all I can do is to search for a initial status (positon, velocity & relative positions) and let the simulator run by itself.

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