Short answers, pulled from the story.
The Titius-Bode law is a formula that predicts the spacing of planets in a planetary system. It takes the sequence 0, 3, 6, 12, 24, 48, 96, adds 4 to each value, and divides by 10 to produce distances in astronomical units. Each value in the sequence after the first is double the previous, so the formula predicts each planet should be roughly twice as far from the Sun as the one before it.
The law is named after Johann Daniel Titius and Johann Elert Bode, but the underlying pattern predates both of them. A similar series appears in a 1715 textbook by D. Gregory, and a version framing planetary distances as a geometric progression with ratio 2 was written by D. Gregory in 1702. Titius, a disciple of the philosopher C. F. von Wolf, likely learned the relation from von Wolf, whose 1723 book contains nearly identical text.
The law successfully anticipated the orbit of Uranus, discovered in 1781, and the position of Ceres in the asteroid belt, found in 1801. When originally published, it also matched the known planets Mercury through Saturn. Neptune, discovered in 1846, was a major failure: the law predicts Neptune at about 38.8 AU, but its actual distance is close to 30 AU.
Neptune's 1846 discovery at the wrong predicted distance dealt the law its most damaging blow, and C. S. Peirce cited it as an example of fallacious reasoning in 1898. The journal Icarus no longer accepts papers attempting to improve the formula. Astrophysicist Alan Boss has stated the pattern is simply a coincidence.
M. A. Blagg, an Oxford astronomer, published a revised formulation in 1913 after analyzing the orbits of planets and the satellite systems of Jupiter, Saturn, and Uranus. She found the progression ratio of 2 used by Titius and Bode was less accurate than a different ratio, and her formula predicted positions for six bodies not yet discovered in 1913 that were later confirmed. Her paper was forgotten until A. E. Roy rediscovered it in 1953.
A generalized Titius-Bode relation was applied to 68 exoplanet systems with four or more known planets, and 96% of those systems adhered to the relation to a similar or greater extent than our Solar System. The analysis predicted 97 potentially undetected planets across those systems, though only 5 candidate planets from that list have since been detected.