Mysterious Numbers From Gold List Approximants

An approximant is the best fractional approximation to a target for a given denominator. Keeping only the approximants that are better than all the previous ones produces the gold list. The gold list of approximants of a real number can exhibit fascinating patterns. This Demonstration lets you examine the gold lists of 12 different numbers, including irrationals like and , and two rational targets. You can choose to see the error terms, which decrease in absolute value. But the mystery is when we (separately) subtract numerators and denominators of successive gold list entries, as shown in the last column. Amazingly enough, most of these appeared previously in the gold list itself