How to find patterns in a series of timestamps
The question can be super difficult if you only have the number of clicks per time-stamp. The reason is that you might find many different recurrent patterns!
For instance certain time periods in year, month or day (according to the functionality of website) may attract a certain temporal pattern of clicks.
But if you have more relevant features the story is different.

After all I will point out some directions and hope it helps:

Autocorrelation may initially give you a good insight about what is happening there.

Then go for Time-Warping methods to define similarity function and seek it along time-series. 

Embedding in high-dimensional spaces disclose the temporal recurrent pattern of a time-series. It's mostly from physics literature and is called State-Space Recunstruction. This method transfers your time-series into a m-dimensional space using this and there you can see the time-series as a dynamic system which may help in detecting recurrent dynamics.