A362504 Number of paths across a regular hexagonal grid with side length n.

1, 11, 291, 8547, 259123, 7997355, 250062515, 7900153283, 251686581987, 8073549437643, 260453164773827, 8441983967058723, 274715571055249171, 8969873019615172459, 293727992832116083539, 9642550297656399476611, 317240324217012764748739

Offset

1,2

Comments

Each side of the hexagonal grid is tiled with n hexagons. A path starts at one vertex and ends at the diametrically opposite vertex. Each step comprises moving one cell down, one cell down-right or one cell up-right (no moving up or left).

Links

Table of n, a(n) for n=1..17.

Matthew Blair, Rigoberto Flórez, and Antara Mukherjee, Honeycombs in the Hosoya Triangle, arXiv:2203.13205 [math.HO], 2022.

Example

For n=2 there are 11 paths that cross diametrically opposite tiles in a honeycomb with 2 hexagons on each side. For n=3 there are 291 such paths.

Prog

(Python)
def paths(x, y, n):
    if (x, y) in [(0, 0), (0, 1), (1, 0)]:
        return 1
    elif x == 0:
        return paths(x, y - 1, n)
    elif y == 0:
        return paths(x - 1, y, n)
    elif y == x + n - 1:
        return paths(x, y - 1, n) + paths(x - 1, y - 1, n)
    elif x == y + n - 1:
        return paths(x - 1, y, n) + paths(x - 1, y - 1, n)
    else:
        return paths(x, y - 1, n) + paths(x - 1, y, n) + paths(x - 1, y - 1, n)
def honeycomb_paths(n):
    return paths(2 * n - 2, 2 * n - 2, n)

Crossrefs

Sequence in context: A064758 A030428 A219099 * A133515 A258191 A350938

Adjacent sequences: A362501 A362502 A362503 * A362505 A362506 A362507

Keyword

nonn,walk

Author

Alexandra S. Kim, Apr 21 2023

Status

approved