A390122 Number of directed cycles in the n X n grid graph, reduced for symmetry (with rotations, reflections, and translations not counted as distinct).

0, 1, 4, 44, 1899, 267976, 112877832

Offset

1,3

Comments

Excludes 0 length cycles; including them increases each term by 1.

Number of undirected cycles gives A394067. Number of directed cycles not reduced for symmetry gives A140517*2.

Links

Table of n, a(n) for n=1..7.

Example

a(3) = 4 (all symmetric and count as 1 each both undirected and directed):
  +>>+>>+    +  +  +    +  +  +    +>>+>>+
  ^     v                          ^     v
  +  +  +    +>>+>>+    +  +>>+    +<<+  +
  ^     v    ^     v       ^  v       ^  v
  +<<+<<+    +<<+<<+    +  +<<+    +  +<<+.
Nonsymmetric cycles count as 2 each when directed, e.g.:
  +  +--+--+      +  +>>+>>+       +  +<<+<<+
     |     |         ^     v          v     ^
  +  +--+  +      +  +<<+  +       +  +>>+  +
        |  |  ->        ^  v  and        v  ^
  +  +  +  +      +  +  +  +       +  +  +  +
        |  |            ^  v             v  ^
  +  +  +--+      +  +  +<<+       +  +  +>>+.

Prog

(PARI) \\ See Hohn link in A394065.

Crossrefs

Cf. A394067, A140517.

Sequence in context: A127635 A134174 A391487 * A296729 A157193 A224784

Adjacent sequences: A390119 A390120 A390121 * A390123 A390124 A390125

Keyword

nonn,more

Author

Charles L. Hohn, Mar 22 2026

Status

approved