|
%I #24 Sep 06 2021 16:08:50
%S 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,
%T 105344,0,0,1046656,3181104
%N 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.
%C 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.
%C 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.
%C As in A010566 all possible paths are counted, including those that are equivalent via rotation and reflection.
%H A. J. Guttmann and A. R. Conway, <a href="http://dx.doi.org/10.1007/PL00013842">Self-Avoiding Walks and Polygons</a>, Annals of Combinatorics 5 (2001) 319-345.
%H Scott R. Shannon, <a href="/A345676/a345676.txt">Images of the closed loops for n = 15</a>. The line lengths in this text file are long so it may need to be downloaded to be viewed correctly.
%e a(1) to a(14) = 0 as no closed-loop paths are possible.
%e 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.
%Y Cf. A347506, A334720, A336265, A337550, A010566, A000217.
%K nonn,walk,more
%O 1,15
%A _Scott R. Shannon_, Sep 04 2021
|
