Conceptual insights and inspirations from experimental and computational mathematics 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?

 A: The Online Encyclopedia of Integer Sequences (OEIS) inspired a large number of theoretical discoveries in mathematics, following the pattern: from computation to numerical data to conjectures to proofs. 
According to the webpage listing Works citing OEIS,

over 6000 people have found it helpful, and that many of these works say things like "This discovery was made with the help of the OEIS".

A: I am not sure if this qualifies, but Mitchell Feigenbaum used
an HP-65 calculator to discover the Feigenbaum constant and used the insights he
gained from his calculations to write
 Quantitative Universality for a Class of Nonlinear Transformations.
In a similar way, Stephen Wolfram's computational experiments
with Cellular Automata led him to work that culminated in his
book A New Kind of Science. These two people were Physicists,
but the results that they came up with are purely mathematical.
A: I'm hardly familiar with this area myself, but the impression I get is that complex dynamics was strongly influenced by computer images of the Julia and Mandelbrot sets. These images revealed a lot of previously unsuspected structure which even from a purely visual perspective is beautiful.
A: 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.
A: Here is just one example: in this 2008 paper, the authors use a cleverly optimized computer search to find two examples of sets of $7$ points in the plane (no three collinear, no four on a circle) so that every pair of points is an integer distance apart. 
This resolved the question, posed by Erdős, of whether such a set of $7$ points exists. 
As Gerry Myerson mentioned in another answer, there are easily thousands of examples in which computation has aided and/or inspired new results. 
A: 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.   
A: I think we could fill pages with examples. How about Lorenz' discovery of chaos from the output of an interrupted computer run? Copying at length from Wikipedia: 

Chaos Theory
  In 1961, Lorenz was using a simple digital computer, a Royal McBee LGP-30, to simulate weather patterns by modeling 12 variables, representing things like temperature and wind speed. He wanted to see a sequence of data again, and to save time he started the simulation in the middle of its course. He did this by entering a printout of the data that corresponded to conditions in the middle of the original simulation. To his surprise, the weather that the machine began to predict was completely different from the previous calculation. The culprit: a rounded decimal number on the computer printout. The computer worked with 6-digit precision, but the printout rounded variables off to a 3-digit number, so a value like 0.506127 printed as 0.506. This difference is tiny, and the consensus at the time would have been that it should have no practical effect. However, Lorenz discovered that small changes in initial conditions produced large changes in long-term outcome.
Lorenz's discovery, which gave its name to Lorenz attractors, showed that even detailed atmospheric modelling cannot, in general, make precise long-term weather predictions. His work on the topic culminated in the publication of his 1963 paper "Deterministic Nonperiodic Flow" in Journal of the Atmospheric Sciences, and with it, the foundation of chaos theory. He states in that paper:

Two states differing by imperceptible amounts may eventually evolve into two considerably different states ... If, then, there is any error whatever in observing the present state—and in any real system such errors seem inevitable—an acceptable prediction of an instantaneous state in the distant future may well be impossible....In view of the inevitable inaccuracy and incompleteness of weather observations, precise very-long-range forecasting would seem to be nonexistent.

His description of the butterfly effect, the idea that small changes can have large consequences, followed in 1969.
Lorenz's insights on deterministic chaos resonated widely starting in the 1970s and 80s, when it spurred new fields of study in virtually every branch of science, from biology to geology to physics. In meteorology, it led to the conclusion that it may be fundamentally impossible to predict weather beyond two or three weeks with a reasonable degree of accuracy. However, the recognition of chaos has led to improvements in weather forecasting, as now forecasters recognize that measurements are imperfect and thus run many simulations starting from slightly different conditions, called ensemble forecasting.
Of the seminal significance of Lorenz's work, Kerry Emanuel, a prominent meteorologist and climate scientist at MIT, has stated:

By showing that certain deterministic systems have formal predictability limits, Ed put the last nail in the coffin of the Cartesian universe and fomented what some have called the third scientific revolution of the 20th century, following on the heels of relativity and quantum physics.

Late in his career, Lorenz began to be recognized with international accolades for the importance of his work on deterministic chaos. In 1983, along with colleague Henry Stommel, he was awarded the Crafoord Prize from the Swedish Academy of Sciences, considered to be nearly equal to a Nobel Prize. He was also awarded the Kyoto Prize for basic sciences in the field of earth and planetary sciences in 1991, the Buys Ballot Award in 2004, and the Tomassoni Award in 2008. In 2018, a short documentary was made about Lorenz's immense scientific legacy on everything from how we predict weather to our understanding of the universe.

A: I don't know if my comment is a contribution for your nice question (I'm an amateur, feel free to add a comment if it doesn't fit with your question), I know an article from the Spanish version of the journal Scientific American that is Investigación y Ciencia, I say the column by Bartolo Luque Juegos matemáticos, the article is Un nuevo patrón en los números primos, Investigación y Ciencia (Julio 2019). I don't know the research article Bartolo Luque and Lucas Lacasa, The First-Digit Frequencies of Prime Numbers and Riemann Zeta Zeros, Proceedings: Mathematical, Physical and Engineering Sciences, Vol. 465, No. 2107 (Jul. 8, 2009), Royal Society. Thus I don't know if it have been inspired by computational search or experimentations, or if there are more reasonings in such article.
