A392623 Number of perfect mazes on an n X n grid, including a discontiguous, directed pair of unit-length entry and exit openings in the border, with rotations and reflections not counted as distinct.

1, 21, 2708, 2702336, 24410499168, 2104896553943040, 1778444003240055248640, 14939769469723143490555084800, 1259088365662049769772300690661574656, 1070947202979258392244674442279023145910272000, 9231770604513665531846104652770226955879223011179905024

Offset

1,2

Comments

Counting the entry/exit pairs as nondirected gives A390017. a(n) = 2*A390017(n) minus 1 for each symmetric case, except for cases which have only 2-way reflection symmetry and both openings intersecting the reflection axis (applies only to odd n), e.g. 3 cases for n = 3:

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Links

Table of n, a(n) for n=1..11.

Example

For a(2) = 21, the 9 nonsymmetric cases counting twice each (once for each entry/exit direction) are:
     _      _      _      _      _
  |_  |  |_     |  _|  |  _   |  _|
   _ _|  |_ _|  |_  |  |_ _|  |_ _
     _      _      _      _
  |   |  |   |  |      |   |
  | |_|  |_| |  |_|_|  |_|_
  and the 3 symmetric cases counting once each (entry/exit directions are congruent through reflection) are:
            _      _
  | | |  |_  |  |  _|
  |_ _|  |  _|  |  _|

Prog

(PARI) vector(11, n, A392623(n)) \\ See Links in A390017 for program code. - Andrew Howroyd, Feb 22 2026

Crossrefs

Cf. A349718, A390017.

Sequence in context: A122801 A099680 A352086 * A184367 A114934 A098375

Adjacent sequences: A392620 A392621 A392622 * A392624 A392625 A392626

Keyword

nonn

Author

Charles L. Hohn, Feb 15 2026

Extensions

a(6) onward from Andrew Howroyd, Feb 22 2026

Status

approved