Publishing a mathematical research article on research which is already done? Recently, I was working on some mathematics (number theory) and try to do that just for a math fun. Prior to that, I did not refer to any literature or any research articles, as I was doing this for fun. But I went in very deep and spent almost 15 days on that as it was getting interesting. 

After one stage, I was able to represent the numbers in a series representation but was not able to find the equation. After a few more days of work, I was able to extract the equation. Now, I can represent that series in one equation. This makes me very curious, as all this I did just by trial and error method and found on my own.

I know that, making equation of some series is not a big task. But, the way I came to that equation is making me think to publish an article on that.

I thought to search on whether this equation is available or not, and as it was expected the research on that has already been done.

In this scenario, can I publish a research article on the topic?
An article is not about you. It is not a reward of effort. In an article you try to communicate to the readers, so it needs to contain something that your readers want to read. If something is already done, then why would I want to read it? Maybe, you have variation, or a special case, or a more general case, that makes your equation or your derivation thereof different and interesting. But if that is not the case, then the answer is simply no.