In Praise of Sequence (Co-)Algebra and its implementation in Haskell

What is Sequence Algebra? This is a question that any teacher or student of mathematics or computer science can engage with. Sequences are in Calculus, Combinatorics, Statistics and Computation. They are foundational, a step up from number arithmetic. Sequence operations are easy to implement from scratch (in Haskell) and afford a wide variety of testing and experimentation. When bits and pieces of sequence algebra are pulled together from the literature, there emerges a claim for status as a substantial pre-analysis topic. Here we set the stage by bringing together a variety of sequence algebra concepts for the first time in one paper. This provides a novel economical overview, intended to invite a broad mathematical audience to cast an eye over the subject. A complete, yet succinct, basic implementation of sequence operations is presented, ready to play with. The implementation also serves as a benchmark for introducing Haskell by mathematical example.