A394959 Number of directed 2n-step 2-dimensional closed self-avoiding paths on square lattice, reduced for symmetry; i.e., where rotations and reflections are not counted as distinct.

0, 1, 1, 3, 9, 36, 157, 758, 3871, 20582, 112701, 631039, 3598161, 20826912, 122107107

Offset

1,4

Comments

Equals 2 * A266549 (undirected paths), minus cases which have vertical, horizontal, or diagonal reflection symmetry, for which the paths in opposite directions are congruent and count as 1 each instead of 2. Nonsymmetric cases and cases with rotation symmetry alone are not congruent in opposite directions and count as 2 each.

Links

Table of n, a(n) for n=1..15.

Example

a(5) = 9: these 3 length 2n=10 figures with reflection symmetry count 1 each (paths in opposite directions are congruent):
  +--+--+--+--+    +--+--+--+    +--+--+--+
  |           |    |        |    |        |
  +--+--+--+--+    +--+  +--+    +        +
                      |  |       |        |
                      +--+       +--+--+--+
  and these 3 figures without reflection symmetry count 2 each (paths in opposite directions are not congruent):
  +--+--+       +--+--+--+    +--+--+--+
  |     |       |        |    |        |
  +--+  +--+    +--+--+  +    +--+     +
     |     |          |  |       |     |
     +--+--+          +--+       +--+--+

Crossrefs

Cf. A266549.

Sequence in context: A350451 A245888 A379034 * A295739 A156016 A376420

Adjacent sequences: A394952 A394953 A394958 * A394960 A394961 A394962

Keyword

nonn,more,walk

Author

Charles L. Hohn, Apr 07 2026

Status

approved