What is the pre-requisite knowledge for generating my own integer sequence?

Question

I've recently come across the On-Line Encyclopedia of Integer Sequences and I'm completely fascinated by it; something about how easy integers are to grasp and yet how complex the sequences are. I find it intriguing and I'd love to be able to contribute to it. Naturally the sequences that are easier to come by have already been submitted and having only studied mathematics up to pre-calculus thus far I feel it is beyond my current abilities to discover something original.

And so my questions are:

1. What would I require asides ingenuity to create/discover an original sequence?
2. What topics would be helpful to study? Algebraic number theory?
3. What topic should I study to build upon sequences & series that I've encountered in pre-calculus?
4. Would I need to be able to programme? Or have a working knowledge of mathematica or another programming language?

Essentially, what knowledge would I require that would allow me to investigate this topic further and create my own sequence? Any books or other resources would also be appreciated.

Thank you.

Answer 1

Reviewing the topics here at MSE is also a good way to provide new interesting series to OEIS. I would suggest you to use the tags of MSE and read all the great questions other users did here, that will give you insights and new ideas. In my case it was so, I have published only two sequences, and both were related with topics I just asked here and later converted into series.

About your specific questions:

What would I require asides ingenuity to create/discover an original sequence?

Once you find a topic that you like, as said above combinatorics, geometry, etc. do not force yourself to find something, just read about the topics you like, learn more, let your brain thinking about it freely. And every time you have an idea, take a paper and write it, and if you can, make a little program and test it.

What topics would be helpful to study? Algebraic number theory?

Imho, any topic you like is fine. I think that there is always a relationship or an idea that can be converted into a sequence. (Initially) I can not imagine a topic unrelated with sequences in one way or another.

Would I need to be able to programme? Or have a working knowledge of mathematica or another programming language?

If you can learn some easy scripting language (Python, PARI\GP, etc.) it will help you to find (or confirm properties, etc.) sequences. It is a powerful tool.

The most important thing is just that interesting ideas come when you are not obsessed with them. When you learn something, the brain requires some time to create new bonds\clusters with the information you assimilate, and the ideas will come fluidly if you take your time to learn about the concepts and play with them. The more you write, test, etc. the more probabilities you have to find something interesting.

Answer 2

You can come up with any sequence you like. If you look around OEIS, you'll find from the ubiquitous Fibonacci numbers to the Look and say sequence.

But consider that they prominently state that they have a huge backlog of proposed sequences, so you should make sure (a) it isn't a variant of something already in there, and (b) is is a really interesting sequence. For both, you might want to ask for opinions here...

Answer 3

Firstly, The submission stack at OEIS is pretty large at the moment. You could help by registering and editing some of the submissions if you feel competent doing that. You might learn a bit in the process. You could also help by finding un-submitted sequences in peer reviewed literature, if you have access to that.

1. Nothing really, Submitting a sequence is very easy. You just need to register, wait for approval, then submit a sequence when you find one.

2. Lots of sequences seem to come from things like combinatorics. You could probably come up with a new sequence by thinking about some sort of lattice path problem for example.

3. It is helpful for enumerating the terms of your sequence if the description is amenable to writing a program. My personal recommendation is python. You might try some problems on projecteuler.net to get you started.