Week One Lattice Animals

Solving the Unsolved

Syllabus

This course will explore the process of mathematical research by examining some apparently simple sequences of whole numbers. The sequences we will consider are related to a variety of problems in combinatorics, probability, number theory, algebra, and geometry. There are many unsolved problems surrounding these sequences, some of them famous and long standing, others only recently posed.

The goal of this course is not to necessarily solve any of these problems (although that would certainly be a wonderful result), but rather to gain experience in exploring unknown territory. This process involves not only the search for solutions, but also the art of asking new questions. Each student will be asked to not only consider the sequences presented in class, but to invent a sequence of their own. Sufficiently interesting sequences may earn a permanent entry in the Online Encyclopedia of Integer Sequences, thereby achieving mathematical immortality.

Week One – Lattice Animals 1, 1, 1, 2, 5, 12, …

Week Two – Derangements 1, 0, 1, 2, 9, 44, …

Week Three – Catalan Numbers 1, 1, 2, 5, 14, 42, …

Week Four – Bell Numbers 1, 1, 2, 5, 15, 52, …

Week Five – Integer Partitions 1, 1, 2, 3, 5, 7, 11, 15, …

Week Six – Trees 1, 1, 1, 1, 2, 3, 6, 11, 23, …

Week Seven – Perfect Primes 2, 3, 5, 7, 13, 17, 19, 31, …

Week Eight – New Sequences 2, 4, 9, 20, 45, …