Inspired by Stanislaw Ulam's spiral of prime numbers on a square grid, a similar approach is used on more mundane integer sequences, with visually surprising results. Starting from single integer multiples displayed on both square and hexagonal grids, which give rise to visually interesting patterns for every positive integer, a simple parameterized indexing algorithm is then used to create a profusion of patterns “in between” the integer multiple ones. This work can serve as the basis for artistic images and animations, and is presented at a level that should be accessible, and perhaps inspirational, to a wide range of STEAM students.