A337550 Number of closed-loop self-avoiding paths of length 4n on a 2D square lattice where no step can be in the same direction as the previous step.

8, 0, 24, 64, 360, 1728, 8624, 43776, 225216, 1173280, 6182704, 32905536, 176657000, 955629920, 5204178360, 28509374976, 157005901896, 868756900608, 4827586102216

Offset

1,1

Comments

See A337353 for the corresponding number of walks.

Only walks with a length of 4n (except for n=2) can create closed loops.

Links

Table of n, a(n) for n=1..19.

A. J. Guttmann and A. R. Conway, Self-Avoiding Walks and Polygons, Annals of Combinatorics 5 (2001) 319-345.

Scott R. Shannon, Text images of the closed-loops for n=1 to n=11.

Example

a(1) = 8. The single walk of length 4 is:
.
+---+
|   |
+---+
.
This can be taken in 8 different ways on a square lattice, giving a total 1*8 = 8.
a(2) = 0 as there is no closed-loop path consisting of 8 steps.
a(3) = 24. There is one walk, ignoring reflection and rotations, with a length of 12. The walk is:
.
    +---+
    |   |
+---+   +---+
|           |
+---+   +---+
    |   |
    +---+
.
This can be walked in 3 different ways if the first steps are right and then upward. This path can be then taken in 8 ways on a square lattice, giving a total number of 3*8 = 24.
a(4) = 64. There is one walk, with indistinct reflections and rotations, with a length of 16. The walk is:
.
        +---+
        |   |
    +---+   +---+
    |           |
+---+       +---+
|           |
+---+   +---+
    |   |
    +---+
.
This can be walked in 8 different ways if the first steps are right and then upward. This path can be then taken in 8 ways on a square lattice, giving a total number of 8*8 = 64.
.
a(5) = 360. There are four walks, with indistinct reflections and rotations, with a length of 20. The walks, and the different ways they can be taken, are:
.
            +---+              +---+
            |   |              |   |
        +---+   +---+      +---+   +---+
        |           |      |           |
    +---+       +---+      +---+       +---+
    |           |              |           |
+---+       +---+          +---+       +---+
|           |              |           |
+---+   +---+              +---+   +---+
    |   |     x 10             |   |     x 20
    +---+                      +---+
        +---+                  +---+
        |   |                  |   |
    +---+   +---+          +---+   +---+
    |           |          |           |
+---+           +---+      +---+   +---+
|                   |          |   |
+---+           +---+      +---+   +---+
    |           |          |           |
    +---+   +---+          +---+   +---+
        |   |    x 5           |   |     x 10
        +---+                  +---+
.
Each of these can be walked in 8 different ways on a square lattice, giving a total number of 8*(10+20+5+10) = 360.
See the attached text file for images of the closed-loops for n=1 to n=11.
      .

Crossrefs

Cf. A337353, A010566, A334720, A036418.

Sequence in context: A167347 A028604 A292531 * A088663 A291681 A011997

Adjacent sequences: A337547 A337548 A337549 * A337551 A337552 A337553

Keyword

nonn,walk,more

Author

Scott R. Shannon, Aug 31 2020

Extensions

a(18)-a(19) from Bert Dobbelaere, Sep 09 2020

Status

approved