TITLE: Conceptual insights and inspirations from experimental and computational mathematics
QUESTION [18 upvotes]: I am interested in whether experiments on computers can help identifying new ideas or concepts in Mathematics. I am not talking about confirming particular conjectures up to certain numbers (for example on the Riemann hypothesis or Collatz conjecture).
I wonder whether there are examples, where results found by computers have been used and understood by mathematicians, who then used this new insight to make real progress in their field?
One example that I recently found is Casey Mann's results on the Heesch Problem called Heesch Numbers of Edge-Marked Polyforms. There, from some exhaustive computational calculation, he has a chapter talking about Interesting examples and observations. While I cannot evaluate the significance of these observations, it goes along the lines what I am searching for:

Do you have examples and literature references with results (new concepts, ideas, insights), that have been inspired by computational search or experimentations?

REPLY [4 votes]: I am quite sure that the paper Using the Logistic Map to Generate Scratching Sounds,
with the amusing abstract

This article presents a mathematical model for generating annoying scratching sounds. Such sounds are generated by frictional motion and
  have been attributed to the chaotic nature of the frequency spectrum
  thereby produced. The proposed model is based on the logistic map and
  is modified to have the stick-slip property of a frictional vibration.
  The resulting sound is similar to that generated by scratching a
  chalkboard or glass plate with the fingernails.

used some sort of computer experiments to figure out what types of mathematical functions sound annoying.

REPLY [2 votes]: There is this recent work "Adventures in Supersingularland" (Arxiv preprint: https://arxiv.org/pdf/1909.07779.pdf) in which several predictions are made about isogeny graphs of supersingular elliptic curves based on experimental data. I am not aware whether the claimed have been proven formally in the meanwhile (that is in the last two months), but I expect the investigation in the preprint to be fertile soil for further (more theoretical) research.