This is a great question and, to my mind, points out the big difficulty with sequence puzzles on a site of this kind. Without the right setting, every sequence puzzle basically reduces to: "what is a cool way to generate this series of numbers?". This is great recipe for getting a "too broad" classification. How do you let the reader know how cool the answer is supposed to be?
There is the "negative" approach. Any series of N numbers can be generated by a polynomial expression of the same order. You can disallow that. You can disallow "multiple sequence" answers (e.g. the odd terms are generated one way and the even terms are generated in another). To my mind this gives the puzzle an artificial feel which detracts from the fun.
I would recommend going in the opposite direction: what makes your sequence puzzle cool? Figure this out, then creating the right setting should be a lot easier. The reason you don't see a lot of this is because cool ideas are hard to come by. I have given this a certain amount of thought and tried a few approaches. Somewhat unsuccessfully here and more successfully here.
The general idea in the first example is that the solution is constrained by the mental capacities of the "creators" in the setting. I think the approach is valid but the execution is flawed. There is a good deal of reading for a puzzle that is rather easy.
In the second series the main gimmick is that it has a fixed number of values. That eliminates a ton of possibilities off the bat. I have an "answer" to that one that talks about puzzle composition and may or may not have some ideas you could use.
How many values should be in the series is a much softer question. Your motto should be: "give until it hurts." You want to give the reader as much as you can without giving away the answer. The second example above uses 37 values. In the Simpletonian puzzle I was very much tempted to leave off the last value but I think in that, at least, I made the right call. It's better to make the puzzle a bit too easy than to leave the reader unsatisfied by a solution that is only marginally better than the alternatives.
Here is a (fairly trivial) example of taking a gimmick and casting it into a story to produce a sequence:
I have two friends one of whom is a chemistry professor and the other a math professor. I invited them over to lunch one day and while I was out of the room getting drinks they started talking. I missed a bit of the start of the conversation but when I came back in, it went as follows:
Who was the mathematician, who was the chemist, and what was Brian's response?
Not a great puzzle, but I'll spoiler tag the answer in case you want to think about it.
Allan is the mathematician. He is giving the digits of pi (3 1 4 1 5 9 2 6). Brian, the chemist, is replying with Avogadro's number (6 0 2 2 1 4 0). The next digit is 8.
The principle is that the sequences mean something. The story gives the reader context to that meaning. It is easy to generate a practically unguessable puzzle by mixing several complicated sequences. I recommend going for the harder task of constructing something that makes the reader work but at the end say: "that was simple, I should have seen it immediately!" The story can either give clues or confirmation.
I hope this helps.