Motzkin Numbers: Circles, Grids, and the Unity of Counting

From a simple circle puzzle with non-crossing chords to Motzkin paths on a grid, the Motzkin numbers weave geometry, combinatorics, and number theory into one elegant sequence. We trace their origins, explore their recurrence and binomial and Catalan connections, and survey the many interpretations (Donaghey & Shapiro’s 13+ counts). Along the way we spotlight surprising prime Motzkin numbers and reflect on what this unity reveals about patterns in mathematics.

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