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

Author

Scott R. Shannon, Sep 04 2021

Keywords

• nonn
• walk
• more

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.

Examples

			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.
		

Links

• 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.

Crossrefs

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