Staircases: a lesson which can work in different ways and yield different payoffs at various levels: pre-algebra, Algebra 1, Algebra 2, Teachers' Mathematics. I link to it here because it can serve as a preview of arithmetic series. (There are detailed notes and some spoilers on pp. 2-3.)

- Lessons from
*Algebra: Themes, Tools, Concepts*by Anita Wah and Henri Picciotto - Arithmetic Sequences and Series | Teacher notes
- Geometric Sequences and Series | Teacher notes
- Nested Squares

Part of the pedagogy behind the above lessons:

- From my Math Education Blog:
- Geometric in More Than One Way!

- An unusual but effective Algebra 2 introduction to sequences:
- Iterating Linear Functions

- Lessons for 11th and 12th grade about triangular, pentagonal, and Fibonacci numbers:
- Stairway to Infinity! Proof by Mathematical Induction

- Directory of material on this site that involves integer sequences, with links to the Online Encyclopedia of Integer Sequences:
- Integer Sequences