The art of algorithmic guessing in $\texttt{gfun}$

The technique of guessing can be very fruitful when dealing with sequences which arise in practice. This holds true especially when guessing is performed algorithmically and efficiently. One highly useful tool for this purpose is the package named $\texttt{gfun}$ in the software Maple. In this text we explore and explain some of $\texttt{gfun}$'s possibilities and illustrate them on two examples from recent mathematical research by the author and his collaborators.