We describe two classes of sequences that give rise to accessible problems for undergraduate research. These problems may be understood with virtually no prerequisites and are well suited for computer-aided investigation. The first sequence is a variation of one introduced by Stephen Wolfram in connection with his study of cellular automata. The second is an exploration of a modified "look and say" sequence that was introduced by Richard Guy and John Conway. For each sequence we discuss the questions that motivated their study and outline the progress that has been made on them by undergraduates. Finally, we suggest several open questions for future student investigation.