The widespread availability of large language models (LLMs), such as ChatGPT, has significantly impacted education, raising both opportunities and challenges. Students can frequently interact with LLM-powered, interactive learning tools, but their usage patterns need to be observed and understood. We introduce StudyChat, a publicly available dataset capturing real-world student interactions with an LLM-powered tutoring chatbot in a semester-long, university-level artificial intelligence (AI) course. We deploy a web application that replicates ChatGPT's core functionalities, and use it to log student interactions with the LLM while working on programming assignments. We collect 16,851 interactions, which we annotate using a dialogue act labeling schema inspired by observed interaction patterns and prior research. We analyze these interactions, highlight usage trends, and analyze how specific student behavior correlates with their course outcome. We find that students who prompt LLMs for conceptual understanding and coding help tend to perform better on assignments and exams. Moreover, students who use LLMs to write reports and circumvent assignment learning objectives have lower outcomes on exams than others. StudyChat serves as a shared resource to facilitate further research on the evolving role of LLMs in education.