There's a probability model called the Pólya urn where you imagine an urns containing numbered balls (colored balls in a typical example, but to draw the comparison with dice we can say they're numbered 1-6), and every time you draw a ball of a certain color, you put back more balls according to some rule. A few probability distributions can be expressed in terms of a Pólya urn, see https://en.wikipedia.org/wiki/P%C3%B3lya_urn_model.

A fair 6-sided die would be an equal number of balls numbered 1-6 and a rule that you simply return the ball you drew. You can get a gambler's fallacy distribution by, say, adding one of every ball that you didn't draw. I read the code as a Pólya urn starting with 1 ball 1-N and doing that on each draw plus reducing the number of balls of the drawn number to 1.

Also related, in 2d space, is the idea of randomly covering the plane in points but getting a spread-out distribution, since uniformity will result in clusters. (If you're moving a small window in any direction and you haven't seen a point in a while, you're "due" to see another one, and vice versa if you just saw a point.) Mike Bostock did a very nice visualization of that here: https://bost.ocks.org/mike/algorithms/