A345676 Number of closed-loop self-avoiding paths on a 2-dimensional square lattice where each path consists of steps with successive lengths equal to the square numbers, from 1 to n^2.

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 32, 0, 0, 0, 368, 264, 0, 0, 1656, 5104, 0, 0, 62016, 105344, 0, 0, 1046656, 3181104

Offset

1,15

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 at each step to the next square number until the step length is n^2. No closed-loop path is possible until n = 15.

Like A334720 and A335305 the only n values that can form closed loop walks are those which correspond to the indices of even triangular numbers. Curiously though n = 16 walks form no closed loops, even though both n = 15 and n = 16 are indices of such numbers.

As in A010566 all possible paths are counted, including those that are equivalent via rotation and reflection.

Links

Table of n, a(n) for n=1..32.

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

Scott R. Shannon, Images of the closed loops for n = 15. The line lengths in this text file are long so it may need to be downloaded to be viewed correctly.

Example

a(1) to a(14) = 0 as no closed-loop paths are possible.
a(15) = 32 as there are four different paths which form closed loops, and each of these can be walked in eight different ways on a 2D square lattice. These walks consist of steps with lengths 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225. See the linked text images.

Crossrefs

Cf. A347506, A334720, A336265, A337550, A010566, A000217.

Sequence in context: A145210 A091308 A023927 * A240253 A057376 A079312

Adjacent sequences: A345673 A345674 A345675 * A345677 A345678 A345679

Keyword

nonn,walk,more

Author

Scott R. Shannon, Sep 04 2021

Status

approved