Publishing a Simple Paper as an Undergraduate First off I apologize if this question does not belong here, I would be happy to hear about any better locations to post this on.
I am a (first year) undergraduate mathematics student, and I recently discovered some interesting properties hidden in certain families of sequences.  I ran these ideas past my math professors, and they said these are interesting ideas and the kind of thing I may want to consider publishing.  I did pretty thorough research through my school's library and I am highly confident at this point that these ideas have not been written on before.  
To give some context into the situation, I am attending a top 15 university, and my professors are all Harvard, Cambridge, Princeton, etc. educated, so I know they know what they are talking about.  With that said, the ideas I discovered -- while interesting -- are really pretty trivial as they can be completely understood with background in Calculus III and some linear algebra, and all my proofs fit onto about 3 pages of LaTeX.   
I would love to go ahead with trying to write up a paper on these ideas and get them published as I think that would be a great experience, however I do not want to run the risk of coming off as an over-confident "crank" type early on and be stained by that.  Would you recommend I keep my cards close to my chest so to speak, or do you think I should put my ideas out there and see what happens?  If I was to pursue publication, I would just submit to undergraduate journals, or maybe something else a little more specialized.  
 A: After writing a manuscript (which it seems you may have already), go through it and revise it a few times until you feel that it is in a polished form. Then you could ask your professors to read it and provide some feedback and revise accordingly. This revision/feedback process will be a good experience for practicing and getting a feel of what the process writing research mathematics is like. Plus, you may come across potential generalizations or other cases you may not have previously considered.
It's important to stress here that you should listen to your professors. They will generally know what journals / media that your paper would be suitable for. For instance, they could advise you on whether it would be worth doing any of the following:


*

*Presenting the material at an undergraduate mathematics conference (e.g. the MAA Meetings).

*Post the article on arXiv

*Publishing the article in an undergraduate journal (e.g. Involve, the Rose-Hulman Undergraduate Mathematics Journal, the Stanford Undergraduate Research Journal)

*Publishing the article in a professional research journal


These could definitely be valuable experiences to get a feel for mathematics research!

Even if you don't publish your current research, the experience is usually more useful in the following two ways than the actual mathematics:


*

*Understanding the research process

*Motivating future mathematical study


Regarding (1), use this process to see if research mathematics is something that you want to pursue! If anything, you'll learn how to present logical arguments cohesively.
Regarding (2), explore the connections of what you are studying to fields of mathematics that you haven't learned yet. Perhaps when looking at generalizations or applications of your result, you'll find that you'll need some deeper mathematical theory X. Use this as motivation to go learn X! Maybe after studying X, you'll have a much better understanding of your previous results when going back to it. Who knows, maybe the ideas from your initial research can be useful for something else much further down the line, regardless of whether it was publishable in its initial form!
A: I encountered the exact same situation myself as an undergraduate, and I would echo robinz16's advice. In my case I ended up publishing it as a Note (short article, typically well under five pages) in Mathematics Magazine. From Wikipedia:

[Mathematics Magazine's] intended audience is teachers of collegiate mathematics, especially at the junior/senior level, and their students. It is explicitly a journal of mathematics rather than pedagogy. Rather than articles in the terse "theorem-proof" style of research journals, it seeks articles which provide a context for the mathematics they deliver, with examples, applications, illustrations, and historical background.

The articles (especially the Notes) are often written by undergraduates and often fall into the "fun and interesting, but not groundbreaking" category. That might be a good fit for your article.
But definitely run it by a trusted professor before you submit it! There are lots of little conventions for formatting/presenting math results (some rather arbitrary) which can only be learned through experience.
A: To those journals already mentioned, I can add College Mathematical Journal published by MAA. But my advise is to ask those professors who approved your result. They must be able to give a good recommendation, after you show them the finished paper. In any case, I recommend you to show your finished paper to a professor.
I have similar experience: I wrote my first paper on my second undergraduate year.
I showed it to my professor, he recommended a (mainstream) journal, and the paper was accepted. Several of my friends mathematicians had a similar experience. 
