Exact Motzkin Number Calculator

Start with one empty path at height zero, then propagate up, down, and level steps while discarding moves below zero. The empty path supplies M(0) = 1 under the displayed indexing convention. Inputs must be whole numbers inside the displayed domain; the page never rounds a decimal into an accepted index. This ties the answer to one explicit convention instead of silently mixing sequence offsets or combinatorial interpretations.

Motzkin paths include horizontal steps, so this count is not interchangeable with a Catalan Dyck-path count. Track a small table by step and height and compare the final zero-height entry. Recompute the displayed recurrence or closed form with the same inputs and compare its previous terms or counting factors. That check supports this bounded result, but it does not transfer the interpretation to a different sequence or counting object.

Path length is capped at 30 for small deterministic state tables and exact output. The bounded state table keeps browser work predictable. Combinatorial and recurrence values can grow rapidly even when the inputs look small. The conservative cap prevents browser stalls and avoids presenting an unsafe floating-point integer as exact; larger work needs arbitrary-precision software and independent resource controls.