The astonishing behavior of recursive sequences
quantamagazine.org
quantamagazine.org
A math professor I had in college, Dave Kelly, had a trick where he'd show a seemingly reasonably-behaved sequence, explain where it came from, then show that the rest of the sequence was absurd -- like, it would be exponential for a while and then zero, or increase arithmetically for a while and then become undefined, etc. I wish I had taped that.
Recursive/fractal sequences are toughest because there's very likely no direct computation. Say your sequence matches with a fractal sequence at only every fifth element, you still have to compute the elements you don't need to get the ones you do need. Sometimes you need a number theory math library to do the computation and that kind of dependency is overkill so now you've got to figure out how recreate just the parts of that math library you need or discover a shortcut (again, using the sequence database.) Invariably you'll discover something new and be able to contribute back.
You build the small-N brute forcing demo, and then the integer sequences (from correct answers, or from how the overall number of solutions grows) may help find the formal math jargon name behind whatever's going on.
For example, Stirling Numbers of the First Kind [0].
[0] https://en.m.wikipedia.org/wiki/Stirling_numbers_of_the_firs...
As always, there is a numberphile video.
A well-known example: 5th Euclid axiom leads to Euclidean geometry, while a different axiom leads to non-euclidean geometries.
Are you talking about the behavior of the floor operator in determining the number of terms in the sum? For k in {1,2,3}, floor(k/2) = 1. So there would only be one term and thus no addition.