Explore the concept of on-sequences in this mathematics lecture. Begin by understanding the limitations of finite sequences and the need for a more modest approach to infinite sequences. Learn about constant on-sequences and their arithmetic, which parallels that of natural numbers. Delve into polynumber (polynomial) on-sequences, including the basic natural number sequence. Examine examples from Sloane's Online Encyclopedia of Integer Sequences, such as the frog-and-toad-hopping-interchange sequence. Gain insights into building a theory of on-sequences step-by-step, visualizing constant on-sequences, and performing arithmetic with polynumber on-sequences. Connect these concepts to real-world applications and expand your understanding of mathematical sequences.