login
A334720
Number of 2-dimensional closed-loop self-avoiding paths on a square lattice where each path consists of steps with incrementing length from 1 to n.
9
0, 0, 0, 0, 0, 0, 8, 24, 0, 0, 40, 112, 0, 0, 1376, 2008, 0, 0, 21720, 60848, 0, 0, 635544, 1517368, 0, 0, 20008456, 46010640, 0, 0, 640819936, 1571759136, 0, 0, 22704325648, 55436103264
OFFSET
1,7
COMMENTS
This sequence gives the number of closed-loop self avoiding walks on a 2D square lattice where the walk starts with a step length of 1 which then increments by 1 after each step up until the step length is n. No closed-loop path is possible until n = 7.
Like A010566 all possible paths are counted, including those that are equivalent via rotation and reflection.
LINKS
A. J. Guttmann and I. G. Enting, The size and number of rings on the square lattice, J. Phys. A 21 (1988), L165-L172.
EXAMPLE
a(1) to a(6) = 0 as no closed-loop is possible.
a(7) = 8 as there is one path which forms a closed loop which can be walked in 8 different ways on a 2D square lattice. The path is:
.
5
*---.---.---.---.---*
| |
. .
| |
. . 4
| |
6 . .
| | 3
. *---.---.---*
| |
. . 2
| |
*---.---.---.---.---.---.---X---*
7 1
.
See the attached link for text images of the closed loops for other n values.
CROSSREFS
KEYWORD
nonn,more,walk
AUTHOR
Scott R. Shannon, May 08 2020
STATUS
approved