We define a card-dealing procedure by an infinite word with the letters U and D: a U-move sends the top card to the bottom of the deck, while a D-move removes the top card. For each pattern, we study five associated objects: the move-count sequence; the dealing triangle, whose entries give the removal rank of each initial position; its inverse, the Josephus triangle; the initial position of the last card removed; and the removal rank of the initial top card. A single round of the procedure gives simultaneous recursions for all five objects. For periodic patterns, we obtain additional row recurrences, and for the under-down shuffle, we express the dealing triangle explicitly in terms of the 2-adic valuation. We collect the resulting OEIS entries, including the 44 sequences introduced in the course of this project.