Grid Paths with Alternating Steps

Sequel to https://puzzlespree.substack.com/p/grid-paths-with-variable-steps

Compute the number of distinct paths in an m x n grid from the top-left to the bottom-right corner, where each move follows a dynamic step size sequence. Starting with any step size from 1 to a given limit, subsequent steps proceed in a round-robin pattern, cycling through the range. Paths must stay within grid boundaries.

Examples:

• count_paths(m=1, n=2, limit=2) returns 2: D1R2, R2D1

  

• count_paths(m=3, n=2, limit=2) returns 2: R2D1D2, D2D1R2

• count_paths(m=3, n=4, limit=3) returns 3: D1D2R3R1, R1D2R3D1, R1R2D3R1

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