The Fibonacci sequence, defined as starting with 1 and 1 and each subsequent number being the sum of the previous two, has far-reaching significance in the natural world. It appears in nautilus shells, the bones in our fingers, and the arrangement of leaves on tree branches. Other sequences, like the Somos sequences and the Göbel sequences, have also captured the interest of mathematicians. The Somos sequences have properties that are surprising and have connections to various areas of mathematics. The Göbel sequences, on the other hand, involve division and produce integer terms for an extended period of time before becoming fractions. The existence of these sequences and their properties fascinate mathematicians and reveal hidden depths in mathematics.
https://www.quantamagazine.org/the-astonishing-behavior-of-recursive-sequences-20231116/