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.

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

Author

Scott R. Shannon, May 08 2020

Keywords

• nonn
• more
• walk

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.

For n = 8, 15, 20, 24, 27, 32, 35, 39, 44, ... = A380867, the path can be a rectangle. The first two cases are illustrated through the "Images" link from Scott R. Shannon. These numbers correspond to triangular numbers T(n) for which there are n1 > n2 > n3 > n4 >= 0 such that T(n) = 2(A+B) for A = T(n1) - T(n2) = T(n3) - T(n4) and B = T(n2) - T(n3). See A380867 for more. - M. F. Hasler, Mar 14 2025

Examples

			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.
		

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.
• Scott R. Shannon, Images of the closed-loops for n=7,8,11,12,15.

Crossrefs

Cf. A010566, A334877, A002931, A334756, A380867.