I like sequences that have non-conventional definitions. For example, there is the very non-equational Look and Say Sequence made famous by Conway:
This starts with the seed value 1, and each successive term is generated by looking at the previous term, saying the numbers appearing there out loud, and writing down the numbers you find yourself saying. For instance, the term ‘111221’ becomes “Three ones, two twos, one one”, which transliterates to ‘312211’.
Despite this crazy generating rule, there is actually a lot of structure to be found in this sequence. Conway found that certain strings of digits, once created, never interacted with those to their left or right again, instead going through an internal ‘life cycle’, growing and changing until it reached a point where it was a string of several such atomic strings joined together; each of these then went off in their own life cycle like some strange numerical mitosis. Conway actually named these atomic strings after the elements, since he found 92 such atomic strings containing the numbers 1, 2, and 3 alone, and two ‘transuranic’ strings for each other natural number.
Conway also found that the ratio of the length of successive terms approaches a constant, and gave a degree-71 polynomial of which this constant is the only real root.
The Look and Say Sequence is surprisingly fruitful, given how non-mathematical its rule seems.